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GCSE Maths · EdexcelChapter 6 · Standard form, indices and surds
GCSE Maths · Standard Form, Indices and Surds · Higher · Lesson 6.8

GCSE Maths: Multiplying, dividing and expanding with surds (Higher)

Part of the GCSE Maths standard form, indices and surds chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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6.8

Multiplying, dividing and expanding with surds

Edexcel 1MA1: N8, A4
Grade 7-9

To multiply surds, multiply the numbers in front and multiply the numbers under the roots: (2√3) × (5√6) = 10√18 = 10 × 3√2 = 30√2. Always simplify at the end. When a surd is multiplied by itself you get a whole number: √3 × √3 = 3, and (2√3)2 = 4 × 3 = 12. To divide, divide the numbers in front and the numbers under the roots: 6√10 ÷ 2√5 = 3√2.

Expanding brackets with surds uses exactly the same method as in algebra: multiply every term in one bracket by every term in the other. The picture shows (2 + √3)(1 + √3) as the area of a rectangle in four parts.

2×1 = 22√3√3×1 = √3√3×√3 = 32√31√3(2 + √3)(1 + √3) = 2 + 2√3 + √3 + 3
The four parts are 2, 2√3, √3 and 3. Adding gives 5 + 3√3.

Remember the special products. (a + √b)2 = a2 + 2a√b + b: do not forget the middle term. And the difference of two squares: (a + √b)(a − √b) = a2 − b. The two middle terms cancel, and the surd disappears. This last result is the key to rationalising the denominator in section 6.9.

Key ideas

  • (a√b) × (c√d) = ac√(bd); then simplify.
  • √a × √a = a; (c√a)2 = c2a.
  • Expand brackets: every term × every term (First, Outer, Inner, Last), then collect like terms.
  • (a + √b)(a − √b) = a2 − b: a whole number.
Worked example 6.8.1
Simplify (2√3) × (5√6).

Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.

Method. Multiply the numbers in front, multiply the numbers under the roots, then simplify.

  1. Multiply the numbers outside the roots. The numbers in front of the roots are multiplied together.
    2 × 5 = 10
  2. Multiply the numbers inside the roots. The rule is √a × √b = √(ab).
    √3 × √6 = √18
  3. Put them together.
    10√18
  4. Simplify the surd if you can. Look for a square number that divides into 18.
    √18 = √9 × 2 = 3√2
    10 × 3√2 = 30√2
  5. Write the answer.
    30√2

Check. Decimals: 3.464 × 12.247 ≈ 42.43, and the answer is about 42.43. ✓

Watch out. Do not multiply the number in front with the number under the root. And √3 × √3 = 3 (a whole number), not √9 × 2.

Worked example 6.8.2
Simplify 6√10 ÷ 2√5.

Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.

Method. Divide the numbers in front, divide the numbers under the roots, then simplify.

  1. Divide the numbers outside the roots. The numbers in front of the roots are divided.
    6 ÷ 2 = 3
  2. Divide the numbers inside the roots. The rule is √a ÷ √b = √(a ÷ b).
    √10 ÷ √5 = √10 ÷ 5 = √2
  3. Put them together and simplify. If the number under the root is a square number, take its root.
    3√2

Check. Check by multiplying back: 3√2 × 2√5 = 6√10, which is 6√10 (simplified). ✓

Watch out. Divide the numbers under the roots, not the roots themselves.

Worked example 6.8.3
Expand and simplify (3 + √2)(4 − √2).

Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.

Method. Expand the brackets by multiplying every term by every term. Simplify each product, then collect like terms.

  1. Write the product. We multiply every term in the first bracket by every term in the second bracket. A way to remember the four products is First, Outer, Inner, Last (FOIL).
    (3 + √2)(4 − √2)
  2. Multiply term by term. Simplify each product. Use √a × √a = a and √a × √b = √(ab). Take care with the signs.
    First: 3 × 4 = 12
    Outer: 3 × (−√2) = −3√2
    Inner: √2 × 4 = 4√2
    Last: √2 × (−√2) = −√4 = −2
  3. Write all the simplified terms in one line. Keep the signs.
    12 − 3√2 + 4√2 − 2
  4. Collect like terms. Add the whole numbers together. Add the like surds together (the same number under the root).
    10 + √2

Check. Decimals: the original is 4.414 × 2.586 ≈ 11.41 and the answer is about 11.41. ✓

Watch out. (2 + √3)2 is not 4 + 3. Always write out both brackets: (2 + √3)(2 + √3). The middle terms do not disappear.

Worked example 6.8.4
Expand and simplify (2 + √3)2.

Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.

Method. Expand the brackets by multiplying every term by every term. Simplify each product, then collect like terms.

  1. Write the product. We multiply every term in the first bracket by every term in the second bracket. A way to remember the four products is First, Outer, Inner, Last (FOIL).
    (2 + √3)(2 + √3)
  2. Multiply term by term. Simplify each product. Use √a × √a = a and √a × √b = √(ab). Take care with the signs.
    First: 2 × 2 = 4
    Outer: 2 × √3 = 2√3
    Inner: √3 × 2 = 2√3
    Last: √3 × √3 = √9 = 3
  3. Write all the simplified terms in one line. Keep the signs.
    4 + 2√3 + 2√3 + 3
  4. Collect like terms. Add the whole numbers together. Add the like surds together (the same number under the root).
    7 + 4√3

Check. Decimals: the original is 3.732 × 3.732 ≈ 13.93 and the answer is about 13.93. ✓

Watch out. (2 + √3)2 is not 4 + 3. Always write out both brackets: (2 + √3)(2 + √3). The middle terms do not disappear.

Worked example 6.8.5
Expand and simplify (4 + √5)(4 − √5).

Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.

Method. Expand the brackets by multiplying every term by every term. Simplify each product, then collect like terms.

  1. Write the product. We multiply every term in the first bracket by every term in the second bracket. A way to remember the four products is First, Outer, Inner, Last (FOIL).
    (4 + √5)(4 − √5)
  2. Multiply term by term. Simplify each product. Use √a × √a = a and √a × √b = √(ab). Take care with the signs.
    First: 4 × 4 = 16
    Outer: 4 × (−√5) = −4√5
    Inner: √5 × 4 = 4√5
    Last: √5 × (−√5) = −√25 = −5
  3. Write all the simplified terms in one line. Keep the signs.
    16 − 4√5 + 4√5 − 5
  4. Collect like terms. Add the whole numbers together. Add the like surds together (the same number under the root).
    11

Check. Decimals: the original is 6.236 × 1.764 ≈ 11.00 and the answer is about 11.00. ✓

Watch out. (2 + √3)2 is not 4 + 3. Always write out both brackets: (2 + √3)(2 + √3). The middle terms do not disappear.

Worked example 6.8.6
Expand and simplify √6(√2 + √3).

Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.

Method. Expand the brackets by multiplying every term by every term. Simplify each product, then collect like terms.

  1. Write the product. We multiply every term in the first bracket by every term in the second bracket.
    √6(√2 + √3)
  2. Multiply term by term. Simplify each product. Use √a × √a = a and √a × √b = √(ab). Take care with the signs.
    √6 × √2 = √12 = √4 × 3 = 2√3
    √6 × √3 = √18 = √9 × 2 = 3√2
  3. Write all the simplified terms in one line. Keep the signs.
    2√3 + 3√2
  4. Collect like terms. Add the whole numbers together. Add the like surds together (the same number under the root).
    3√2 + 2√3

Check. Decimals: the original is 2.449 × 3.146 ≈ 7.71 and the answer is about 7.71. ✓

Watch out. (2 + √3)2 is not 4 + 3. Always write out both brackets: (2 + √3)(2 + √3). The middle terms do not disappear.

Watch out

(2 + √3)2 is not 4 + 3. Write the two brackets out: (2 + √3)(2 + √3) = 4 + 2√3 + 2√3 + 3 = 7 + 4√3. Do not lose the middle terms.

Exercise 6P  Multiplying, dividing and expanding with surds

Non-calculator
1
FSimplify each product.
(a)√2 × √8
(b)√3 × √5
(c)2√3 × 5√6
(d)3√2 × 4√2
(e)√6 × √15
(f)4√5 × 2√5
2
FSimplify each division.
(a)6√10 ÷ 2√5
(b)√24 ÷ √6
(c)8√18 ÷ 2√2
(d)√45 ÷ √5
3
FWork out each value.
(a)(√7)2
(b)(3√5)2
(c)(2√3)2
4
FExpand and simplify.
(a)√3(2 + √3)
(b)√5(√5 − 2)
(c)2√2(3 + √2)
(d)√6(√2 + √3)
5
FExpand and simplify.
(a)(3 + √2)(4 − √2)
(b)(1 + √3)(2 + √3)
(c)(√5 + 2)(√5 − 1)
(d)(2 + √3)(2 + √3)
(e)(1 − √2)(1 − √2)
(f)(4 + √5)(4 − √5)
6
PSA rectangle has length (2 + √3) cm and width (1 + √3) cm. Show that its area is (5 + 3√3) cm2.
2×1 = 22√3√3×1 = √3√3×√3 = 32√31√3(2 + √3)(1 + √3) = 2 + 2√3 + √3 + 3
7
PSA rectangle has length (4 + √2) cm and width (4 − √2) cm. Work out its exact area and its exact perimeter.
(a)area
(b)perimeter
8
RShow that (√5 + 2)(√5 − 2) = 1.
9
RShow that (√8 + √2)2 = 18.
10
RIs (3 + √2)2 equal to 11? Explain.
11
RShow that (a + √b)(a − √b) is always a whole number when a and b are whole numbers.
12
Rx = 2 + √3. Show that x2 − 4x + 1 = 0.
13
RShow that (√3 + √2)2 − (√3 − √2)2 = 4√6.

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