GCSE Maths: Squares, cubes, roots and powers (Foundation)

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To square a number, multiply it by itself.

1. Write it as a multiplication. Squaring means multiplying the number by itself.
   4² = 4 × 4
2. Work it out. Use your times tables.
   4 × 4 = 16

Check. The last digit of 4² comes from 4 × 4 = 16, so the answer should end in 6. ✓

Watch out. 4² means 4 × 4, not 4 × 2.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To square a number, multiply it by itself.

1. Write it as a multiplication. Squaring means multiplying the number by itself.
   7² = 7 × 7
2. Work it out. Use your times tables.
   7 × 7 = 49

Check. The last digit of 7² comes from 7 × 7 = 49, so the answer should end in 9. ✓

Watch out. 7² means 7 × 7, not 7 × 2.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To square a number, multiply it by itself.

1. Write it as a multiplication. Squaring means multiplying the number by itself.
   9² = 9 × 9
2. Work it out. Use your times tables.
   9 × 9 = 81

Check. The last digit of 9² comes from 9 × 9 = 81, so the answer should end in 1. ✓

Watch out. 9² means 9 × 9, not 9 × 2.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To square a number, multiply it by itself.

1. Write it as a multiplication. Squaring means multiplying the number by itself.
   11² = 11 × 11
2. Split one of the numbers into tens and units and multiply each part. This is long multiplication done in two easy steps: 11 × 10 and 11 × 1.
   11 × 10 = 110
   11 × 1 = 11
3. Add the two parts. Add the results of the two multiplications.
   110 + 11 = 121

Check. The last digit of 11² comes from 1 × 1 = 1, so the answer should end in 1. ✓

Watch out. 11² means 11 × 11, not 11 × 2.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To square a number, multiply it by itself.

1. Write it as a multiplication. Squaring means multiplying the number by itself.
   12² = 12 × 12
2. Split one of the numbers into tens and units and multiply each part. This is long multiplication done in two easy steps: 12 × 10 and 12 × 2.
   12 × 10 = 120
   12 × 2 = 24
3. Add the two parts. Add the results of the two multiplications.
   120 + 24 = 144

Check. The last digit of 12² comes from 2 × 2 = 4, so the answer should end in 4. ✓

Watch out. 12² means 12 × 12, not 12 × 2.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To square a number, multiply it by itself.

1. Write it as a multiplication. Squaring means multiplying the number by itself.
   15² = 15 × 15
2. Split one of the numbers into tens and units and multiply each part. This is long multiplication done in two easy steps: 15 × 10 and 15 × 5.
   15 × 10 = 150
   15 × 5 = 75
3. Add the two parts. Add the results of the two multiplications.
   150 + 75 = 225

Check. The last digit of 15² comes from 5 × 5 = 25, so the answer should end in 5. ✓

Watch out. 15² means 15 × 15, not 15 × 2.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To cube a number, multiply three copies of it together.

1. Write it as a multiplication. Cubing means multiplying three copies of the number.
   2³ = 2 × 2 × 2
2. Do the first two numbers first. Multiply in stages so that each stage is easy.
   2 × 2 = 4
3. Multiply by the third number. Now multiply the answer by the last copy.
   4 × 2 = 8

Check. Go the other way: the cube root of 8 should be 2, and 2 × 2 × 2 = 8. ✓

Watch out. 2³ means 2 × 2 × 2, not 2 × 3.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To cube a number, multiply three copies of it together.

1. Write it as a multiplication. Cubing means multiplying three copies of the number.
   3³ = 3 × 3 × 3
2. Do the first two numbers first. Multiply in stages so that each stage is easy.
   3 × 3 = 9
3. Multiply by the third number. Now multiply the answer by the last copy.
   9 × 3 = 27

Check. Go the other way: the cube root of 27 should be 3, and 3 × 3 × 3 = 27. ✓

Watch out. 3³ means 3 × 3 × 3, not 3 × 3.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To cube a number, multiply three copies of it together.

1. Write it as a multiplication. Cubing means multiplying three copies of the number.
   4³ = 4 × 4 × 4
2. Do the first two numbers first. Multiply in stages so that each stage is easy.
   4 × 4 = 16
3. Multiply by the third number. Now multiply the answer by the last copy.
   16 × 4 = 64

Check. Go the other way: the cube root of 64 should be 4, and 4 × 4 × 4 = 64. ✓

Watch out. 4³ means 4 × 4 × 4, not 4 × 3.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To cube a number, multiply three copies of it together.

1. Write it as a multiplication. Cubing means multiplying three copies of the number.
   5³ = 5 × 5 × 5
2. Do the first two numbers first. Multiply in stages so that each stage is easy.
   5 × 5 = 25
3. Multiply by the third number. Now multiply the answer by the last copy.
   25 × 5 = 125

Check. Go the other way: the cube root of 125 should be 5, and 5 × 5 × 5 = 125. ✓

Watch out. 5³ means 5 × 5 × 5, not 5 × 3.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To cube a number, multiply three copies of it together.

1. Write it as a multiplication. Cubing means multiplying three copies of the number.
   10³ = 10 × 10 × 10
2. Do the first two numbers first. Multiply in stages so that each stage is easy.
   10 × 10 = 100
3. Multiply by the third number. Now multiply the answer by the last copy.
   100 × 10 = 1000

Check. Go the other way: the cube root of 1000 should be 10, and 10 × 10 × 10 = 1000. ✓

Watch out. 10³ means 10 × 10 × 10, not 10 × 3.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To cube a number, multiply three copies of it together.

1. Write it as a multiplication. Cubing means multiplying three copies of the number.
   7³ = 7 × 7 × 7
2. Do the first two numbers first. Multiply in stages so that each stage is easy.
   7 × 7 = 49
3. Multiply by the third number. Now multiply the answer by the last copy.
   49 × 7 = 343

Check. Go the other way: the cube root of 343 should be 7, and 7 × 7 × 7 = 343. ✓

Watch out. 7³ means 7 × 7 × 7, not 7 × 3.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A square root is the number that, multiplied by itself, gives the number inside the root.

1. Think: which number multiplied by itself gives 25? A square root undoes a square.
   We want ? × ? = 25
2. Look at the squares you know that are close to the number. Use your list of square numbers to home in on the answer.
   3² = 9
   4² = 16
   5² = 25
   6² = 36
   7² = 49
3. Pick the one that matches. Only one square equals the number.
   5 × 5 = 25, so √25 = 5

Check. Square the answer: 5² = 25. ✓

Watch out. √25 means the positive number 5. (The equation x² = 25 has two answers, 5 and −5, but the √ sign gives only the positive one.)

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A square root is the number that, multiplied by itself, gives the number inside the root.

1. Think: which number multiplied by itself gives 64? A square root undoes a square.
   We want ? × ? = 64
2. Look at the squares you know that are close to the number. Use your list of square numbers to home in on the answer.
   6² = 36
   7² = 49
   8² = 64
   9² = 81
   10² = 100
3. Pick the one that matches. Only one square equals the number.
   8 × 8 = 64, so √64 = 8

Check. Square the answer: 8² = 64. ✓

Watch out. √64 means the positive number 8. (The equation x² = 64 has two answers, 8 and −8, but the √ sign gives only the positive one.)

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A square root is the number that, multiplied by itself, gives the number inside the root.

1. Think: which number multiplied by itself gives 81? A square root undoes a square.
   We want ? × ? = 81
2. Look at the squares you know that are close to the number. Use your list of square numbers to home in on the answer.
   7² = 49
   8² = 64
   9² = 81
   10² = 100
   11² = 121
3. Pick the one that matches. Only one square equals the number.
   9 × 9 = 81, so √81 = 9

Check. Square the answer: 9² = 81. ✓

Watch out. √81 means the positive number 9. (The equation x² = 81 has two answers, 9 and −9, but the √ sign gives only the positive one.)

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A square root is the number that, multiplied by itself, gives the number inside the root.

1. Think: which number multiplied by itself gives 121? A square root undoes a square.
   We want ? × ? = 121
2. Look at the squares you know that are close to the number. Use your list of square numbers to home in on the answer.
   9² = 81
   10² = 100
   11² = 121
   12² = 144
   13² = 169
3. Pick the one that matches. Only one square equals the number.
   11 × 11 = 121, so √121 = 11

Check. Square the answer: 11² = 121. ✓

Watch out. √121 means the positive number 11. (The equation x² = 121 has two answers, 11 and −11, but the √ sign gives only the positive one.)

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A square root is the number that, multiplied by itself, gives the number inside the root.

1. Think: which number multiplied by itself gives 144? A square root undoes a square.
   We want ? × ? = 144
2. Look at the squares you know that are close to the number. Use your list of square numbers to home in on the answer.
   10² = 100
   11² = 121
   12² = 144
   13² = 169
   14² = 196
3. Pick the one that matches. Only one square equals the number.
   12 × 12 = 144, so √144 = 12

Check. Square the answer: 12² = 144. ✓

Watch out. √144 means the positive number 12. (The equation x² = 144 has two answers, 12 and −12, but the √ sign gives only the positive one.)

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A square root is the number that, multiplied by itself, gives the number inside the root.

1. Think: which number multiplied by itself gives 196? A square root undoes a square.
   We want ? × ? = 196
2. Look at the squares you know that are close to the number. Use your list of square numbers to home in on the answer.
   12² = 144
   13² = 169
   14² = 196
   15² = 225
   16² = 256
3. Pick the one that matches. Only one square equals the number.
   14 × 14 = 196, so √196 = 14

Check. Square the answer: 14² = 196. ✓

Watch out. √196 means the positive number 14. (The equation x² = 196 has two answers, 14 and −14, but the √ sign gives only the positive one.)

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A cube root is the number that, multiplied by itself three times, gives the number.

1. Think: which number multiplied by itself three times gives 8? A cube root undoes a cube.
   We want ? × ? × ? = 8
2. Look at the cubes you know. Use your list of cube numbers.
   1³ = 1
   2³ = 8
   3³ = 27
   4³ = 64
3. Pick the one that matches. Only one cube equals the number.
   2 × 2 × 2 = 8, so the cube root of 8 is 2

Check. Cube the answer: 2³ = 8. ✓

Watch out. The cube root of 8 is 2, not 8 ÷ 3.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A cube root is the number that, multiplied by itself three times, gives the number.

1. Think: which number multiplied by itself three times gives 27? A cube root undoes a cube.
   We want ? × ? × ? = 27
2. Look at the cubes you know. Use your list of cube numbers.
   2³ = 8
   3³ = 27
   4³ = 64
   5³ = 125
3. Pick the one that matches. Only one cube equals the number.
   3 × 3 × 3 = 27, so the cube root of 27 is 3

Check. Cube the answer: 3³ = 27. ✓

Watch out. The cube root of 27 is 3, not 27 ÷ 3.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A cube root is the number that, multiplied by itself three times, gives the number.

1. Think: which number multiplied by itself three times gives 64? A cube root undoes a cube.
   We want ? × ? × ? = 64
2. Look at the cubes you know. Use your list of cube numbers.
   3³ = 27
   4³ = 64
   5³ = 125
   6³ = 216
3. Pick the one that matches. Only one cube equals the number.
   4 × 4 × 4 = 64, so the cube root of 64 is 4

Check. Cube the answer: 4³ = 64. ✓

Watch out. The cube root of 64 is 4, not 64 ÷ 3.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A cube root is the number that, multiplied by itself three times, gives the number.

1. Think: which number multiplied by itself three times gives 125? A cube root undoes a cube.
   We want ? × ? × ? = 125
2. Look at the cubes you know. Use your list of cube numbers.
   4³ = 64
   5³ = 125
   6³ = 216
   7³ = 343
3. Pick the one that matches. Only one cube equals the number.
   5 × 5 × 5 = 125, so the cube root of 125 is 5

Check. Cube the answer: 5³ = 125. ✓

Watch out. The cube root of 125 is 5, not 125 ÷ 3.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A cube root is the number that, multiplied by itself three times, gives the number.

1. Think: which number multiplied by itself three times gives 1000? A cube root undoes a cube.
   We want ? × ? × ? = 1000
2. Look at the cubes you know. Use your list of cube numbers.
   9³ = 729
   10³ = 1000
   11³ = 1331
   12³ = 1728
3. Pick the one that matches. Only one cube equals the number.
   10 × 10 × 10 = 1000, so the cube root of 1000 is 10

Check. Cube the answer: 10³ = 1000. ✓

Watch out. The cube root of 1000 is 10, not 1000 ÷ 3.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A cube root is the number that, multiplied by itself three times, gives the number.

1. Think: which number multiplied by itself three times gives 343? A cube root undoes a cube.
   We want ? × ? × ? = 343
2. Look at the cubes you know. Use your list of cube numbers.
   6³ = 216
   7³ = 343
   8³ = 512
   9³ = 729
3. Pick the one that matches. Only one cube equals the number.
   7 × 7 × 7 = 343, so the cube root of 343 is 7

Check. Cube the answer: 7³ = 343. ✓

Watch out. The cube root of 343 is 7, not 343 ÷ 3.

Key words. In 3⁴, the 3 is the base and the 4 is the power (or index). It means 3 × 3 × 3 × 3.

Method. A power tells you how many copies of the base to multiply together.

1. Write out the multiplication. The power 5 tells us to write 2 5 times and multiply.
   2⁵ = 2 × 2 × 2 × 2 × 2
2. Multiply one step at a time. Keep a running total so that every step is a small multiplication.
   2 × 2 = 4
   4 × 2 = 8
   8 × 2 = 16
   16 × 2 = 32

Check. There are 5 copies of 2 in the multiplication, which matches the power 5. ✓

Watch out. 2⁵ is not 2 × 5.

Key words. In 3⁴, the 3 is the base and the 4 is the power (or index). It means 3 × 3 × 3 × 3.

Method. A power tells you how many copies of the base to multiply together.

1. Write out the multiplication. The power 4 tells us to write 3 4 times and multiply.
   3⁴ = 3 × 3 × 3 × 3
2. Multiply one step at a time. Keep a running total so that every step is a small multiplication.
   3 × 3 = 9
   9 × 3 = 27
   27 × 3 = 81

Check. There are 4 copies of 3 in the multiplication, which matches the power 4. ✓

Watch out. 3⁴ is not 3 × 4.

Key words. In 3⁴, the 3 is the base and the 4 is the power (or index). It means 3 × 3 × 3 × 3.

Method. A power tells you how many copies of the base to multiply together.

1. Write out the multiplication. The power 3 tells us to write 5 3 times and multiply.
   5³ = 5 × 5 × 5
2. Multiply one step at a time. Keep a running total so that every step is a small multiplication.
   5 × 5 = 25
   25 × 5 = 125

Check. There are 3 copies of 5 in the multiplication, which matches the power 3. ✓

Watch out. 5³ is not 5 × 3.

Key words. In 3⁴, the 3 is the base and the 4 is the power (or index). It means 3 × 3 × 3 × 3.

Method. A power tells you how many copies of the base to multiply together.

1. Write out the multiplication. The power 4 tells us to write 4 4 times and multiply.
   4⁴ = 4 × 4 × 4 × 4
2. Multiply one step at a time. Keep a running total so that every step is a small multiplication.
   4 × 4 = 16
   16 × 4 = 64
   64 × 4 = 256

Check. There are 4 copies of 4 in the multiplication, which matches the power 4. ✓

Watch out. 4⁴ is not 4 × 4.

Key words. In 3⁴, the 3 is the base and the 4 is the power (or index). It means 3 × 3 × 3 × 3.

Method. A power tells you how many copies of the base to multiply together.

1. Write out the multiplication. The power 8 tells us to write 2 8 times and multiply.
   2⁸ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
2. Multiply one step at a time. Keep a running total so that every step is a small multiplication.
   2 × 2 = 4
   4 × 2 = 8
   8 × 2 = 16
   16 × 2 = 32
   32 × 2 = 64
   64 × 2 = 128
   128 × 2 = 256

Check. There are 8 copies of 2 in the multiplication, which matches the power 8. ✓

Watch out. 2⁸ is not 2 × 8.

Key words. In 3⁴, the 3 is the base and the 4 is the power (or index). It means 3 × 3 × 3 × 3.

Method. A power tells you how many copies of the base to multiply together.

1. Write out the multiplication. The power 5 tells us to write 10 5 times and multiply.
   10⁵ = 10 × 10 × 10 × 10 × 10
2. Multiply one step at a time. Keep a running total so that every step is a small multiplication.
   10 × 10 = 100
   100 × 10 = 1000
   1000 × 10 = 10,000
   10,000 × 10 = 100,000

Check. There are 5 copies of 10 in the multiplication, which matches the power 5. ✓

Watch out. 10⁵ is not 10 × 5.

Key words. A power of a number is that number multiplied by itself a certain number of times.

Method. Keep multiplying by 2 until you reach 32, and count how many 2s you used.

1. Start with the base and multiply by it again and again. Each line is one more power of the base.
   2¹ = 2
   2² = 4
   2³ = 8
   2⁴ = 16
   2⁵ = 32
2. Read off the power. 32 appears after 5 multiplications, so the power is 5.
   32 = 2⁵
3. Check by counting. The number of 2s multiplied is the index.
   2 × 2 × 2 × 2 × 2 = 32

Check. 2⁵ = 32. ✓

Key words. A power of a number is that number multiplied by itself a certain number of times.

Method. Keep multiplying by 3 until you reach 81, and count how many 3s you used.

1. Start with the base and multiply by it again and again. Each line is one more power of the base.
   3¹ = 3
   3² = 9
   3³ = 27
   3⁴ = 81
2. Read off the power. 81 appears after 4 multiplications, so the power is 4.
   81 = 3⁴
3. Check by counting. The number of 3s multiplied is the index.
   3 × 3 × 3 × 3 = 81

Check. 3⁴ = 81. ✓

Key words. A power of a number is that number multiplied by itself a certain number of times.

Method. Keep multiplying by 5 until you reach 125, and count how many 5s you used.

1. Start with the base and multiply by it again and again. Each line is one more power of the base.
   5¹ = 5
   5² = 25
   5³ = 125
2. Read off the power. 125 appears after 3 multiplications, so the power is 3.
   125 = 5³
3. Check by counting. The number of 5s multiplied is the index.
   5 × 5 × 5 = 125

Check. 5³ = 125. ✓

Key words. A power of a number is that number multiplied by itself a certain number of times.

Method. Keep multiplying by 4 until you reach 256, and count how many 4s you used.

1. Start with the base and multiply by it again and again. Each line is one more power of the base.
   4¹ = 4
   4² = 16
   4³ = 64
   4⁴ = 256
2. Read off the power. 256 appears after 4 multiplications, so the power is 4.
   256 = 4⁴
3. Check by counting. The number of 4s multiplied is the index.
   4 × 4 × 4 × 4 = 256

Check. 4⁴ = 256. ✓

Key words. A power of a number is that number multiplied by itself a certain number of times.

Method. Keep multiplying by 3 until you reach 729, and count how many 3s you used.

1. Start with the base and multiply by it again and again. Each line is one more power of the base.
   3¹ = 3
   3² = 9
   3³ = 27
   3⁴ = 81
   3⁵ = 243
   3⁶ = 729
2. Read off the power. 729 appears after 6 multiplications, so the power is 6.
   729 = 3⁶
3. Check by counting. The number of 3s multiplied is the index.
   3 × 3 × 3 × 3 × 3 × 3 = 729

Check. 3⁶ = 729. ✓

Key words. A power of a number is that number multiplied by itself a certain number of times.

Method. Keep multiplying by 2 until you reach 1024, and count how many 2s you used.

1. Start with the base and multiply by it again and again. Each line is one more power of the base.
   2¹ = 2
   2² = 4
   2³ = 8
   2⁴ = 16
   2⁵ = 32
   2⁶ = 64
   2⁷ = 128
   2⁸ = 256
   2⁹ = 512
   2¹⁰ = 1024
2. Read off the power. 1024 appears after 10 multiplications, so the power is 10.
   1024 = 2¹⁰
3. Check by counting. The number of 2s multiplied is the index.
   2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024

Check. 2¹⁰ = 1024. ✓

Key words. BIDMAS is the order for working out: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Method. Use BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

1. Write the calculation and decide what must be done first. Spot the brackets and the powers or roots.
   7 + 3²
2. Powers come before adding. In BIDMAS, Indices (powers and roots) are done before Addition.
   3² = 9
3. Now add. Add the result to the other number.
   7 + 9 = 16

Check. Work through the order again step by step, or use a calculator with brackets, to confirm 16. ✓

Watch out. Do not add first: (7 + 3)² = 100 is a different calculation.

Key words. BIDMAS is the order for working out: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Method. Use BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

1. Write the calculation and decide what must be done first. Spot the brackets and the powers or roots.
   (4 + 3)²
2. Brackets first. Work out the bracket before doing the power.
   4 + 3 = 7
3. Now square the result. The power applies to everything in the bracket.
   7² = 49

Check. Work through the order again step by step, or use a calculator with brackets, to confirm 49. ✓

Watch out. (4 + 3)² is not 4² + 3² = 25.

Key words. BIDMAS is the order for working out: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Method. Use BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

1. Write the calculation and decide what must be done first. Spot the brackets and the powers or roots.
   5 × 4²
2. Powers come before multiplying. The power applies only to the number it touches.
   4² = 16
3. Now multiply. Multiply by the other number.
   5 × 16 = 80

Check. Work through the order again step by step, or use a calculator with brackets, to confirm 80. ✓

Watch out. The 2 only squares the 4. 5 × 4² is not (5 × 4)² = 400.

Key words. BIDMAS is the order for working out: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Method. Use BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

1. Write the calculation and decide what must be done first. Spot the brackets and the powers or roots.
   4³ − 6²
2. Powers first. Do both powers before the subtraction.
   4³ = 64
   6² = 36
3. Now subtract. Subtract the second answer from the first.
   64 − 36 = 28

Check. Work through the order again step by step, or use a calculator with brackets, to confirm 28. ✓

Watch out. Cube means three copies, square means two copies. Do not mix them up.

Key words. BIDMAS is the order for working out: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Method. Use BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

1. Write the calculation and decide what must be done first. Spot the brackets and the powers or roots.
   √49 + 2³
2. Roots and powers first. Both are Indices in BIDMAS and are done before adding.
   √49 = 7
   2³ = 8
3. Now add. Add the two answers.
   7 + 8 = 15

Check. Work through the order again step by step, or use a calculator with brackets, to confirm 15. ✓

Watch out. √49 means the square root of 49, not 49 ÷ 2.

Key words. BIDMAS is the order for working out: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Method. Use BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

1. Write the calculation and decide what must be done first. Spot the brackets and the powers or roots.
   √(8² + 15²)
2. Work out the powers inside the root. The root sign acts like a bracket: do the working underneath it first.
   8² = 64
   15² = 225
3. Add inside the root. Add the two squares.
   64 + 225 = 289
4. Take the square root. Now find the number that squares to give the total.
   √289 = 17

Check. Work through the order again step by step, or use a calculator with brackets, to confirm 17. ✓

Watch out. You cannot split it: √(8² + 15²) is not 8 + 15 = 23.

Key words. BIDMAS is the order for working out: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Method. Use BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

1. Write the calculation and decide what must be done first. Spot the brackets and the powers or roots.
   (−4)²
2. The bracket means the minus sign is squared too. (−4)² means (−4) × (−4).
   (−4) × (−4)
3. Multiply. Two negatives make a positive. A negative times a negative is positive.
   = 16

Check. Work through the order again step by step, or use a calculator with brackets, to confirm 16. ✓

Watch out. Compare −4² = −16: with no bracket only the 4 is squared.

Key words. BIDMAS is the order for working out: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Method. Use BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

1. Write the calculation and decide what must be done first. Spot the brackets and the powers or roots.
   −4²
2. Square the number first. The power only touches the 4. The minus sign is applied afterwards.
   4² = 16
3. Now apply the minus sign. Put the minus sign in front of the answer.
   −16

Check. Work through the order again step by step, or use a calculator with brackets, to confirm −16. ✓

Watch out. Compare (−4)² = 16: with a bracket the minus sign is squared too.

Key words. BIDMAS is the order for working out: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Method. Use BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

1. Write the calculation and decide what must be done first. Spot the brackets and the powers or roots.
   (−5)²
2. The bracket means the minus sign is squared too. (−5)² means (−5) × (−5).
   (−5) × (−5)
3. Multiply. Two negatives make a positive. A negative times a negative is positive.
   = 25

Check. Work through the order again step by step, or use a calculator with brackets, to confirm 25. ✓

Watch out. Compare −5² = −25: with no bracket only the 5 is squared.

Key words. BIDMAS is the order for working out: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Method. Use BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

1. Write the calculation and decide what must be done first. Spot the brackets and the powers or roots.
   −5²
2. Square the number first. The power only touches the 5. The minus sign is applied afterwards.
   5² = 25
3. Now apply the minus sign. Put the minus sign in front of the answer.
   −25

Check. Work through the order again step by step, or use a calculator with brackets, to confirm −25. ✓

Watch out. Compare (−5)² = 25: with a bracket the minus sign is squared too.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To square a decimal, square the digits as whole numbers, then count the decimal places.

1. Ignore the decimal point for now and use the whole number. 0.3 is 3 tenths. Work with 3 first.
   3² = 3 × 3 = 9
2. Count the decimal places. 0.3 has 1 decimal place. We multiply 2 copies of it, so the answer has 2 decimal places in total.
   1 decimal place × 2 = 2 decimal places
3. Put the decimal point back. Start with 9 and put the point so that there are 2 digits after it. Add zeros at the front if needed.
   (0.3)² = 0.09

Check. 0.3 is less than 1, so its square is smaller than 0.3. 0.09 < 0.3. ✓

Watch out. Squaring a number between 0 and 1 gives a smaller number: 0.3² = 0.09, not 0.9.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To square a decimal, square the digits as whole numbers, then count the decimal places.

1. Ignore the decimal point for now and use the whole number. 0.5 is 5 tenths. Work with 5 first.
   5² = 5 × 5 = 25
2. Count the decimal places. 0.5 has 1 decimal place. We multiply 2 copies of it, so the answer has 2 decimal places in total.
   1 decimal place × 2 = 2 decimal places
3. Put the decimal point back. Start with 25 and put the point so that there are 2 digits after it. Add zeros at the front if needed.
   (0.5)² = 0.25

Check. 0.5 is less than 1, so its square is smaller than 0.5. 0.25 < 0.5. ✓

Watch out. Squaring a number between 0 and 1 gives a smaller number: 0.3² = 0.09, not 0.9.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To square a decimal, square the digits as whole numbers, then count the decimal places.

1. Ignore the decimal point for now and use the whole number. 1.2 is 12 tenths. Work with 12 first.
   12² = 12 × 12 = 144
2. Count the decimal places. 1.2 has 1 decimal place. We multiply 2 copies of it, so the answer has 2 decimal places in total.
   1 decimal place × 2 = 2 decimal places
3. Put the decimal point back. Start with 144 and put the point so that there are 2 digits after it. Add zeros at the front if needed.
   (1.2)² = 1.44

Check. Estimate: 1.2 is close to 1, and 1² = 1, which is near 1.44. ✓

Watch out. Squaring a number between 0 and 1 gives a smaller number: 0.3² = 0.09, not 0.9.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To cube a decimal, cube the digits as whole numbers, then count the decimal places.

1. Ignore the decimal point for now and use the whole number. 0.2 is 2 tenths. Work with 2 first.
   2³ = 2 × 2 × 2 = 8
2. Count the decimal places. 0.2 has 1 decimal place. We multiply 3 copies of it, so the answer has 3 decimal places in total.
   1 decimal place × 3 = 3 decimal places
3. Put the decimal point back. Start with 8 and put the point so that there are 3 digits after it. Add zeros at the front if needed.
   (0.2)³ = 0.008

Check. 0.2 is less than 1, so its cube is smaller than 0.2. 0.008 < 0.2. ✓

Watch out. Squaring a number between 0 and 1 gives a smaller number: 0.3² = 0.09, not 0.9.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To square a decimal, square the digits as whole numbers, then count the decimal places.

1. Ignore the decimal point for now and use the whole number. 1.5 is 15 tenths. Work with 15 first.
   15² = 15 × 15 = 225
2. Count the decimal places. 1.5 has 1 decimal place. We multiply 2 copies of it, so the answer has 2 decimal places in total.
   1 decimal place × 2 = 2 decimal places
3. Put the decimal point back. Start with 225 and put the point so that there are 2 digits after it. Add zeros at the front if needed.
   (1.5)² = 2.25

Check. Estimate: 1.5 is close to 2, and 2² = 4, which is near 2.25. ✓

Watch out. Squaring a number between 0 and 1 gives a smaller number: 0.3² = 0.09, not 0.9.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To square a decimal, square the digits as whole numbers, then count the decimal places.

1. Ignore the decimal point for now and use the whole number. 1.1 is 11 tenths. Work with 11 first.
   11² = 11 × 11 = 121
2. Count the decimal places. 1.1 has 1 decimal place. We multiply 2 copies of it, so the answer has 2 decimal places in total.
   1 decimal place × 2 = 2 decimal places
3. Put the decimal point back. Start with 121 and put the point so that there are 2 digits after it. Add zeros at the front if needed.
   (1.1)² = 1.21

Check. Estimate: 1.1 is close to 1, and 1² = 1, which is near 1.21. ✓

Watch out. Squaring a number between 0 and 1 gives a smaller number: 0.3² = 0.09, not 0.9.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. Volume of a cube = edge³, so the edge is the cube root of the volume.

1. Write what we know. A cube has three equal edges multiplied together to give the volume.
   edge × edge × edge = 125
2. Find the cube root.
   5 × 5 × 5 = 125, so the edge is 5 cm

Check. 5³ = 125. ✓

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A cube has 6 identical square faces. Find the area of one face and multiply by 6.

1. Find the area of one face. Each face is a square of side 5 cm.
   5 × 5 = 25 cm²
2. Multiply by 6. There are six faces.
   6 × 25 = 150 cm²

Check. 150 ÷ 6 = 25 = 5². ✓

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. Work out the value of each power, then compare them.

1. Work out 2⁸. Multiply the base by itself the number of times in the power.
   2⁸ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256
2. Work out 8². Do the same for the other power.
   8² = 8 × 8 = 64
3. Compare. The bigger value is the bigger number.
   256 > 64, so 2⁸ is bigger.

Check. Powers can be surprising: the bigger base does not always give the bigger answer. ✓

Watch out. 2⁸ and 8² are usually different.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. Work out the value of each power, then compare them.

1. Work out 3⁴. Multiply the base by itself the number of times in the power.
   3⁴ = 3 × 3 × 3 × 3 = 81
2. Work out 4³. Do the same for the other power.
   4³ = 4 × 4 × 4 = 64
3. Compare. The bigger value is the bigger number.
   81 > 64, so 3⁴ is bigger.

Check. Powers can be surprising: the bigger base does not always give the bigger answer. ✓

Watch out. 3⁴ and 4³ are usually different.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. Work out the value of each power, then compare them.

1. Work out 5³. Multiply the base by itself the number of times in the power.
   5³ = 5 × 5 × 5 = 125
2. Work out 3⁵. Do the same for the other power.
   3⁵ = 3 × 3 × 3 × 3 × 3 = 243
3. Compare. The bigger value is the bigger number.
   125 < 243, so 5³ is smaller.

Check. Powers can be surprising: the bigger base does not always give the bigger answer. ✓

Watch out. 5³ and 3⁵ are usually different.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. Find which two square numbers it lies between.

1. Find the two square numbers that are closest to the number. A square number is a whole number multiplied by itself.
   7² = 49 and 8² = 64
2. Compare. The number is a square only if it equals one of them.
   50 is between 49 and 64
3. Decide.
   No, 50 is between 7² and 8².

Check. √50 = 7.071..., which is not a whole number. ✓

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. Find which two square numbers it lies between.

1. Find the two square numbers that are closest to the number. A square number is a whole number multiplied by itself.
   11² = 121 and 12² = 144
2. Compare. The number is a square only if it equals one of them.
   121 = 121
3. Decide.
   Yes, 121 = 11².

Check. √121 = 11.000. A whole number. ✓

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. Find which two square numbers it lies between.

1. Find the two square numbers that are closest to the number. A square number is a whole number multiplied by itself.
   14² = 196 and 15² = 225
2. Compare. The number is a square only if it equals one of them.
   200 is between 196 and 225
3. Decide.
   No, 200 is between 14² and 15².

Check. √200 = 14.142..., which is not a whole number. ✓

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. Find which two square numbers it lies between.

1. Find the two square numbers that are closest to the number. A square number is a whole number multiplied by itself.
   31² = 961 and 32² = 1024
2. Compare. The number is a square only if it equals one of them.
   1000 is between 961 and 1024
3. Decide.
   No, 1000 is between 31² and 32².

Check. √1000 = 31.623..., which is not a whole number. ✓

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A square root is the number that, multiplied by itself, gives the number inside the root.

1. Think: which number multiplied by itself gives 49? A square root undoes a square.
   We want ? × ? = 49
2. Look at the squares you know that are close to the number. Use your list of square numbers to home in on the answer.
   5² = 25
   6² = 36
   7² = 49
   8² = 64
   9² = 81
3. Pick the one that matches. Only one square equals the number.
   7 × 7 = 49, so √49 = 7

Check. Square the answer: 7² = 49. ✓

Watch out. √49 means the positive number 7. (The equation x² = 49 has two answers, 7 and −7, but the √ sign gives only the positive one.)

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. Find every number that squares to 49. Both a positive and a negative number work.

1. Take the square root of both sides. We need numbers that multiply by themselves to give 49.
   x = √49 or x = −√49
2. Work out the root. 7 × 7 = 49 and (−7) × (−7) = 49.
   x = 7 or x = −7

Check. 7² = 49 and (−7)² = 49. ✓

Watch out. A square root sign gives only the positive answer, but an equation can have two solutions.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. Compare what each question asks.

1. Part (a) asks for one value. The symbol √ always means the positive root.
   √49 = 7
2. Part (b) asks for all values of x. A negative number squared is positive too.
   7 and −7
3. Explain.
   The two answers differ because √ is defined as positive only, while the equation has two solutions.

Check. (−7)² = 49, so −7 is a solution of the equation. ✓