GCSE Maths: Multiplying and dividing terms, and the laws of indices (Foundation)

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   4x × 3
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 4 × 3 = 12
   x: add the powers 1 = 1
   So the top is 12x
3. Write the answer. Put the number first, then the letters.
   12x

Check. Test with numbers: put x = 2 into the question and into the answer. The question gives 24 and the answer gives 24. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   2a × 5b
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 2 × 5 = 10
   a: add the powers 1 = 1
   b: add the powers 1 = 1
   So the top is 10ab
3. Write the answer. Put the number first, then the letters.
   10ab

Check. Test with numbers: put a = 2, b = 3 into the question and into the answer. The question gives 60 and the answer gives 60. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   x × x
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 1 × 1 = 1
   x: add the powers 1 + 1 = 2
   So the top is x²
3. Write the answer. Put the number first, then the letters.
   x²

Check. Test with numbers: put x = 2 into the question and into the answer. The question gives 4 and the answer gives 4. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   3y × 2y
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 3 × 2 = 6
   y: add the powers 1 + 1 = 2
   So the top is 6y²
3. Write the answer. Put the number first, then the letters.
   6y²

Check. Test with numbers: put y = 2 into the question and into the answer. The question gives 24 and the answer gives 24. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   5m × m²
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 5 × 1 = 5
   m: add the powers 1 + 2 = 3
   So the top is 5m³
3. Write the answer. Put the number first, then the letters.
   5m³

Check. Test with numbers: put m = 2 into the question and into the answer. The question gives 40 and the answer gives 40. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   ab × ab
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 1 × 1 = 1
   a: add the powers 1 + 1 = 2
   b: add the powers 1 + 1 = 2
   So the top is a²b²
3. Write the answer. Put the number first, then the letters.
   a²b²

Check. Test with numbers: put a = 2, b = 3 into the question and into the answer. The question gives 36 and the answer gives 36. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   x³ × x⁴
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 1 × 1 = 1
   x: add the powers 3 + 4 = 7
   So the top is x⁷
3. Write the answer. Put the number first, then the letters.
   x⁷

Check. Test with numbers: put x = 2 into the question and into the answer. The question gives 128 and the answer gives 128. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   a⁵ ÷ a²
2. Now divide: divide the numbers, and for each letter subtract the powers. Dividing by a term is the same as cancelling. x⁵ ÷ x² = (x × x × x × x × x) ÷ (x × x) = x³ (the law a^m ÷ aⁿ = a^m−n). Divide the numbers as you would divide any numbers.
   Numbers: 1 ÷ 1 = 1
   a: subtract the powers 5 − 2 = 3
3. Write the answer. Put the number first, then the letters.
   a³

Check. Test with numbers: put a = 2 into the question and into the answer. The question gives 8 and the answer gives 8. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   (y²)³
2. Deal with any power outside a bracket first. A power outside a bracket applies to everything inside it: the number and every letter. For example (2x³)² means 2x³ × 2x³. The number is raised to the power, and each letter's own power is multiplied by it (the law (a^m)ⁿ = a^mn).
   (y²)³ → y: 2 × 3 = 6 → y⁶
3. Write the answer. Put the number first, then the letters.
   y⁶

Check. Test with numbers: put y = 2 into the question and into the answer. The question gives 64 and the answer gives 64. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   p × p × p
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 1 × 1 × 1 = 1
   p: add the powers 1 + 1 + 1 = 3
   So the top is p³
3. Write the answer. Put the number first, then the letters.
   p³

Check. Test with numbers: put p = 2 into the question and into the answer. The question gives 8 and the answer gives 8. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   m⁶ ÷ m
2. Now divide: divide the numbers, and for each letter subtract the powers. Dividing by a term is the same as cancelling. x⁵ ÷ x² = (x × x × x × x × x) ÷ (x × x) = x³ (the law a^m ÷ aⁿ = a^m−n). Divide the numbers as you would divide any numbers.
   Numbers: 1 ÷ 1 = 1
   m: subtract the powers 6 − 1 = 5
3. Write the answer. Put the number first, then the letters.
   m⁵

Check. Test with numbers: put m = 2 into the question and into the answer. The question gives 32 and the answer gives 32. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   k⁸ × k
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 1 × 1 = 1
   k: add the powers 8 + 1 = 9
   So the top is k⁹
3. Write the answer. Put the number first, then the letters.
   k⁹

Check. Test with numbers: put k = 2 into the question and into the answer. The question gives 512 and the answer gives 512. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   3x² × 2x⁴
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 3 × 2 = 6
   x: add the powers 2 + 4 = 6
   So the top is 6x⁶
3. Write the answer. Put the number first, then the letters.
   6x⁶

Check. Test with numbers: put x = 2 into the question and into the answer. The question gives 384 and the answer gives 384. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   4a³ × 5a
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 4 × 5 = 20
   a: add the powers 3 + 1 = 4
   So the top is 20a⁴
3. Write the answer. Put the number first, then the letters.
   20a⁴

Check. Test with numbers: put a = 2 into the question and into the answer. The question gives 320 and the answer gives 320. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   6b × b²
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 6 × 1 = 6
   b: add the powers 1 + 2 = 3
   So the top is 6b³
3. Write the answer. Put the number first, then the letters.
   6b³

Check. Test with numbers: put b = 2 into the question and into the answer. The question gives 48 and the answer gives 48. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   2y³ × 7y²
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 2 × 7 = 14
   y: add the powers 3 + 2 = 5
   So the top is 14y⁵
3. Write the answer. Put the number first, then the letters.
   14y⁵

Check. Test with numbers: put y = 2 into the question and into the answer. The question gives 448 and the answer gives 448. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   12x ÷ 3
2. Now divide: divide the numbers, and for each letter subtract the powers. Dividing by a term is the same as cancelling. x⁵ ÷ x² = (x × x × x × x × x) ÷ (x × x) = x³ (the law a^m ÷ aⁿ = a^m−n). Divide the numbers as you would divide any numbers.
   Numbers: 12 ÷ 3 = 4
   x: subtract the powers 1 − 0 = 1
3. Write the answer. Put the number first, then the letters.
   4x

Check. Test with numbers: put x = 2 into the question and into the answer. The question gives 8 and the answer gives 8. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   20a² ÷ 4a
2. Now divide: divide the numbers, and for each letter subtract the powers. Dividing by a term is the same as cancelling. x⁵ ÷ x² = (x × x × x × x × x) ÷ (x × x) = x³ (the law a^m ÷ aⁿ = a^m−n). Divide the numbers as you would divide any numbers.
   Numbers: 20 ÷ 4 = 5
   a: subtract the powers 2 − 1 = 1
3. Write the answer. Put the number first, then the letters.
   5a

Check. Test with numbers: put a = 2 into the question and into the answer. The question gives 10 and the answer gives 10. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   15y³ ÷ 5y
2. Now divide: divide the numbers, and for each letter subtract the powers. Dividing by a term is the same as cancelling. x⁵ ÷ x² = (x × x × x × x × x) ÷ (x × x) = x³ (the law a^m ÷ aⁿ = a^m−n). Divide the numbers as you would divide any numbers.
   Numbers: 15 ÷ 5 = 3
   y: subtract the powers 3 − 1 = 2
3. Write the answer. Put the number first, then the letters.
   3y²

Check. Test with numbers: put y = 2 into the question and into the answer. The question gives 12 and the answer gives 12. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   18p⁴ ÷ 6p²
2. Now divide: divide the numbers, and for each letter subtract the powers. Dividing by a term is the same as cancelling. x⁵ ÷ x² = (x × x × x × x × x) ÷ (x × x) = x³ (the law a^m ÷ aⁿ = a^m−n). Divide the numbers as you would divide any numbers.
   Numbers: 18 ÷ 6 = 3
   p: subtract the powers 4 − 2 = 2
3. Write the answer. Put the number first, then the letters.
   3p²

Check. Test with numbers: put p = 2 into the question and into the answer. The question gives 12 and the answer gives 12. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   8m ÷ 2m
2. Now divide: divide the numbers, and for each letter subtract the powers. Dividing by a term is the same as cancelling. x⁵ ÷ x² = (x × x × x × x × x) ÷ (x × x) = x³ (the law a^m ÷ aⁿ = a^m−n). Divide the numbers as you would divide any numbers.
   Numbers: 8 ÷ 2 = 4
   m: subtract the powers 1 − 1 = 0. A power of 0 means the letter has cancelled completely, because a⁰ = 1.
3. Write the answer. Put the number first, then the letters.
   4

Check. Test with numbers: put m = 2 into the question and into the answer. The question gives 4 and the answer gives 4. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   9t² ÷ 3t²
2. Now divide: divide the numbers, and for each letter subtract the powers. Dividing by a term is the same as cancelling. x⁵ ÷ x² = (x × x × x × x × x) ÷ (x × x) = x³ (the law a^m ÷ aⁿ = a^m−n). Divide the numbers as you would divide any numbers.
   Numbers: 9 ÷ 3 = 3
   t: subtract the powers 2 − 2 = 0. A power of 0 means the letter has cancelled completely, because a⁰ = 1.
3. Write the answer. Put the number first, then the letters.
   3

Check. Test with numbers: put t = 2 into the question and into the answer. The question gives 3 and the answer gives 3. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   3ab × 2a
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 3 × 2 = 6
   a: add the powers 1 + 1 = 2
   b: add the powers 1 = 1
   So the top is 6a²b
3. Write the answer. Put the number first, then the letters.
   6a²b

Check. Test with numbers: put a = 2, b = 3 into the question and into the answer. The question gives 72 and the answer gives 72. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   4xy × 5x
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 4 × 5 = 20
   x: add the powers 1 + 1 = 2
   y: add the powers 1 = 1
   So the top is 20x²y
3. Write the answer. Put the number first, then the letters.
   20x²y

Check. Test with numbers: put x = 2, y = 3 into the question and into the answer. The question gives 240 and the answer gives 240. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   2pq × 3pq
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 2 × 3 = 6
   p: add the powers 1 + 1 = 2
   q: add the powers 1 + 1 = 2
   So the top is 6p²q²
3. Write the answer. Put the number first, then the letters.
   6p²q²

Check. Test with numbers: put p = 2, q = 3 into the question and into the answer. The question gives 216 and the answer gives 216. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   a²b × ab²
2. Multiply the terms: first the numbers, then each letter. Multiply the numbers (coefficients) together. For each letter, add the powers, because x² × x³ = (x × x) × (x × x × x) = x⁵ (the law a^m × aⁿ = a^m+n). A letter written on its own has power 1.
   Numbers: 1 × 1 = 1
   a: add the powers 2 + 1 = 3
   b: add the powers 1 + 2 = 3
   So the top is a³b³
3. Write the answer. Put the number first, then the letters.
   a³b³

Check. Test with numbers: put a = 2, b = 3 into the question and into the answer. The question gives 216 and the answer gives 216. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   (x²)³
2. Deal with any power outside a bracket first. A power outside a bracket applies to everything inside it: the number and every letter. For example (2x³)² means 2x³ × 2x³. The number is raised to the power, and each letter's own power is multiplied by it (the law (a^m)ⁿ = a^mn).
   (x²)³ → x: 2 × 3 = 6 → x⁶
3. Write the answer. Put the number first, then the letters.
   x⁶

Check. Test with numbers: put x = 2 into the question and into the answer. The question gives 64 and the answer gives 64. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   (2a)³
2. Deal with any power outside a bracket first. A power outside a bracket applies to everything inside it: the number and every letter. For example (2x³)² means 2x³ × 2x³. The number is raised to the power, and each letter's own power is multiplied by it (the law (a^m)ⁿ = a^mn).
   (2a)³ → number: 2³ = 8; a: 1 × 3 = 3 → 8a³
3. Write the answer. Put the number first, then the letters.
   8a³

Check. Test with numbers: put a = 2 into the question and into the answer. The question gives 64 and the answer gives 64. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   (3b²)²
2. Deal with any power outside a bracket first. A power outside a bracket applies to everything inside it: the number and every letter. For example (2x³)² means 2x³ × 2x³. The number is raised to the power, and each letter's own power is multiplied by it (the law (a^m)ⁿ = a^mn).
   (3b²)² → number: 3² = 9; b: 2 × 2 = 4 → 9b⁴
3. Write the answer. Put the number first, then the letters.
   9b⁴

Check. Test with numbers: put b = 2 into the question and into the answer. The question gives 144 and the answer gives 144. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   (5x)²
2. Deal with any power outside a bracket first. A power outside a bracket applies to everything inside it: the number and every letter. For example (2x³)² means 2x³ × 2x³. The number is raised to the power, and each letter's own power is multiplied by it (the law (a^m)ⁿ = a^mn).
   (5x)² → number: 5² = 25; x: 1 × 2 = 2 → 25x²
3. Write the answer. Put the number first, then the letters.
   25x²

Check. Test with numbers: put x = 2 into the question and into the answer. The question gives 100 and the answer gives 100. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   (ab)⁴
2. Deal with any power outside a bracket first. A power outside a bracket applies to everything inside it: the number and every letter. For example (2x³)² means 2x³ × 2x³. The number is raised to the power, and each letter's own power is multiplied by it (the law (a^m)ⁿ = a^mn).
   (ab)⁴ → a: 1 × 4 = 4; b: 1 × 4 = 4 → a⁴b⁴
3. Write the answer. Put the number first, then the letters.
   a⁴b⁴

Check. Test with numbers: put a = 2, b = 3 into the question and into the answer. The question gives 1296 and the answer gives 1296. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   (2y³)⁴
2. Deal with any power outside a bracket first. A power outside a bracket applies to everything inside it: the number and every letter. For example (2x³)² means 2x³ × 2x³. The number is raised to the power, and each letter's own power is multiplied by it (the law (a^m)ⁿ = a^mn).
   (2y³)⁴ → number: 2⁴ = 16; y: 3 × 4 = 12 → 16y¹²
3. Write the answer. Put the number first, then the letters.
   16y¹²

Check. Test with numbers: put y = 2 into the question and into the answer. The question gives 65536 and the answer gives 65536. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   6x²3x
2. Now divide: divide the numbers, and for each letter subtract the powers. Dividing by a term is the same as cancelling. x⁵ ÷ x² = (x × x × x × x × x) ÷ (x × x) = x³ (the law a^m ÷ aⁿ = a^m−n). Divide the numbers as you would divide any numbers.
   Numbers: 6 ÷ 3 = 2
   x: subtract the powers 2 − 1 = 1
3. Write the answer. Put the number first, then the letters.
   2x

Check. Test with numbers: put x = 2 into the question and into the answer. The question gives 4 and the answer gives 4. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   10a³b5ab
2. Now divide: divide the numbers, and for each letter subtract the powers. Dividing by a term is the same as cancelling. x⁵ ÷ x² = (x × x × x × x × x) ÷ (x × x) = x³ (the law a^m ÷ aⁿ = a^m−n). Divide the numbers as you would divide any numbers.
   Numbers: 10 ÷ 5 = 2
   a: subtract the powers 3 − 1 = 2
   b: subtract the powers 1 − 1 = 0. A power of 0 means the letter has cancelled completely, because a⁰ = 1.
3. Write the answer. Put the number first, then the letters.
   2a²

Check. Test with numbers: put a = 2, b = 3 into the question and into the answer. The question gives 8 and the answer gives 8. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   12xy²4y
2. Now divide: divide the numbers, and for each letter subtract the powers. Dividing by a term is the same as cancelling. x⁵ ÷ x² = (x × x × x × x × x) ÷ (x × x) = x³ (the law a^m ÷ aⁿ = a^m−n). Divide the numbers as you would divide any numbers.
   Numbers: 12 ÷ 4 = 3
   x: subtract the powers 1 − 0 = 1
   y: subtract the powers 2 − 1 = 1
3. Write the answer. Put the number first, then the letters.
   3xy

Check. Test with numbers: put x = 2, y = 3 into the question and into the answer. The question gives 18 and the answer gives 18. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Key words. The power or index (plural indices) tells you how many copies of the letter are multiplied together: x³ = x × x × x. The coefficient is the number in front of the letters.

Method. To multiply terms, multiply the numbers and add the powers of each letter. To divide, divide the numbers and subtract the powers. A power outside a bracket multiplies every power inside.

1. Write the question. We are going to simplify one product or quotient of terms.
   15p³q²3pq
2. Now divide: divide the numbers, and for each letter subtract the powers. Dividing by a term is the same as cancelling. x⁵ ÷ x² = (x × x × x × x × x) ÷ (x × x) = x³ (the law a^m ÷ aⁿ = a^m−n). Divide the numbers as you would divide any numbers.
   Numbers: 15 ÷ 3 = 5
   p: subtract the powers 3 − 1 = 2
   q: subtract the powers 2 − 1 = 1
3. Write the answer. Put the number first, then the letters.
   5p²q

Check. Test with numbers: put p = 2, q = 3 into the question and into the answer. The question gives 60 and the answer gives 60. They are the same. ✓

Watch out. Add the powers when you multiply (x² × x³ = x⁵, not x⁶). Multiply the powers when there is a power of a power. And only the powers change: 3x × 2x = 6x², the 3 and 2 are multiplied, not added.

Method. When multiplying, the powers add.

1. Write an equation for the powers. ? + 4 = 9.
   ? + 4 = 9
2. Solve. Subtract 4 from both sides.
   ? = 9 − 4 = 5

Check. x⁵ × x⁴ = x⁵⁺⁴ = x⁹. ✓

Method. When dividing, the powers subtract.

1. Write an equation for the powers. 7 − ? = 2.
   7 − ? = 2
2. Solve. The missing number is what you take away from 7 to leave 2.
   ? = 7 − 2 = 5

Check. x⁷ ÷ x⁵ = x². ✓

Method. For a power of a power the powers multiply.

1. Write an equation for the powers. ? × 3 = 12.
   ? × 3 = 12
2. Solve. Divide both sides by 3.
   ? = 12 ÷ 3 = 4

Check. (x⁴)³ = x^4×3 = x¹². ✓

Method. Only the x powers matter here, since 3 × 2 = 6 already matches. Add the powers.

1. Write an equation for the powers. ? + 2 = 5.
   ? + 2 = 5
2. Solve.
   ? = 5 − 2 = 3

Check. 3x³ × 2x² = 6x⁵. ✓

Method. Any number (except 0) raised to the power 0 is 1. Here is why.

1. Divide a power by itself. 5³ ÷ 5³ = 125 ÷ 125 = 1.
   5³ ÷ 5³ = 1
2. Use the subtraction law for indices. The same division gives 5³⁻³.
   5³ ÷ 5³ = 5⁰
3. Put them together. Both are the same calculation, so they are equal.
   5⁰ = 1

Check. The same works for any other power: 5² ÷ 5² = 25 ÷ 25 = 1. ✓

Method. Any letter (not zero) to the power 0 is 1.

1. Divide a power by itself. y⁴ ÷ y⁴ = 1 because anything divided by itself is 1.
   y⁴ ÷ y⁴ = 1
2. Use the subtraction law.
   y⁴⁻⁴ = y⁰, so y⁰ = 1

Check. Try y = 3: 3⁴ ÷ 3⁴ = 81 ÷ 81 = 1. ✓

Method. The power 0 only applies to the letter next to it, not the 6.

1. Replace x⁰ by 1. The index 0 belongs to x only.
   6x⁰ = 6 × 1
2. Multiply.
   6 × 1 = 6

Check. If x = 5, x⁰ = 1, so 6 × 1 = 6. ✓

Watch out. 6x⁰ is 6, but (6x)⁰ is 1.

Method. Everything inside the bracket is raised to the power 0.

1. Raise the whole bracket to the power 0. Any non-zero quantity to the power 0 is 1.
   (3a)⁰ = 1
2. Say why. It is (3a)² ÷ (3a)² = 1.
   1

Check. If a = 2: (3 × 2)⁰ = 6⁰ = 1. ✓