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# GCSE Maths | Surds
- URL: https://www.a-level-maths-tutor.com/gcse-maths-surds/
- Published: 2026-07-29T14:33:51.000Z
- Updated: 2026-07-29T14:35:54.000Z
- Author: Jaisul Naik
- Tags: GCSE Maths

Square numbers: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$

Cube numbers: $1, 8, 27, 64, 125$

Powers of 4: $1, 16, 81$

**Worked Example**

$(\\sqrt{10})^2 = 10$

## Question 1

**a)** $(\\sqrt{15})^2$

**b)** $(\\sqrt{2})^2$

**c)** $(\\sqrt{20})^2$

**d)** $(\\sqrt{11})^2$

**e)** $(\\sqrt{31})^2$

**f)** $(\\sqrt{8})^2$

**g)** $(\\sqrt{45})^2$

**h)** $(\\sqrt{99})^2$

**i)** $(\\sqrt{12})^2$

**j)** $(\\sqrt{18})^2$

Show Answers 

**a)** $15$ **b)** $2$ **c)** $20$ **d)** $11$ **e)** $31$

**f)** $8$ **g)** $45$ **h)** $99$ **i)** $12$ **j)** $18$

**Worked Example**

$(\\sqrt{2})^3 = 2\\sqrt{2}$

$(\\sqrt{2})^4 = 4$

## Question 2

**a)** $(\\sqrt{7})^3$

**b)** $(\\sqrt{11})^3$

**c)** $(\\sqrt{5})^4$

**d)** $(\\sqrt{6})^4$

**e)** $(\\sqrt{2})^5$

**f)** $(\\sqrt{3})^5$

**g)** $(\\sqrt{10})^5$

**h)** $(\\sqrt{2})^6$

**i)** $(\\sqrt{5})^6$

**j)** $(\\sqrt{3})^7$

Show Answers 

**a)** $7\\sqrt{7}$ **b)** $11\\sqrt{11}$ **c)** $25$ **d)** $36$ **e)** $4\\sqrt{2}$

**f)** $9\\sqrt{3}$ **g)** $100\\sqrt{10}$ **h)** $8$ **i)** $125$ **j)** $27\\sqrt{3}$

$\\sqrt{a} \\times \\sqrt{b} = \\sqrt{ab}$

**Worked Example**

$3\\sqrt{2} \\times 4\\sqrt{5} = 12\\sqrt{10}$

## Question 3

**a)** $2\\sqrt{5} \\times 3\\sqrt{7}$

**b)** $4\\sqrt{2} \\times 5\\sqrt{3}$

**c)** $6\\sqrt{11} \\times 2\\sqrt{5}$

**d)** $3\\sqrt{2} \\times 7\\sqrt{13}$

**e)** $5\\sqrt{3} \\times 2\\sqrt{10}$

**f)** $8\\sqrt{7} \\times \\sqrt{2}$

**g)** $4\\sqrt{11} \\times 3\\sqrt{3}$

**h)** $9\\sqrt{5} \\times 2\\sqrt{6}$

**i)** $10\\sqrt{3} \\times 3\\sqrt{11}$

**j)** $7\\sqrt{2} \\times 6\\sqrt{17}$

Show Answers 

**a)** $6\\sqrt{35}$ **b)** $20\\sqrt{6}$ **c)** $12\\sqrt{55}$ **d)** $21\\sqrt{26}$ **e)** $10\\sqrt{30}$

**f)** $8\\sqrt{14}$ **g)** $12\\sqrt{33}$ **h)** $18\\sqrt{30}$ **i)** $30\\sqrt{33}$ **j)** $42\\sqrt{34}$

*\[Video: "Surds (Part 1) Simplifying" goes here\]*

To simplify a surd, find the largest square-number factor and split it out.

**Worked Example**

$\\sqrt{72} = \\sqrt{36 \\times 2} = 6\\sqrt{2}$

## Question 4

**a)** $\\sqrt{8}$

**b)** $\\sqrt{75}$

**c)** $\\sqrt{20}$

**d)** $\\sqrt{32}$

**e)** $\\sqrt{48}$

**f)** $\\sqrt{200}$

**g)** $\\sqrt{300}$

**h)** $\\sqrt{80}$

**i)** $\\sqrt{50}$

**j)** $\\sqrt{98}$

Show Answers 

**a)** $2\\sqrt{2}$ **b)** $5\\sqrt{3}$ **c)** $2\\sqrt{5}$ **d)** $4\\sqrt{2}$ **e)** $4\\sqrt{3}$

**f)** $10\\sqrt{2}$ **g)** $10\\sqrt{3}$ **h)** $4\\sqrt{5}$ **i)** $5\\sqrt{2}$ **j)** $7\\sqrt{2}$

## Question 5

Multiply and simplify each of the following.

**a)** $2\\sqrt{3} \\times 4\\sqrt{5}$

**b)** $5\\sqrt{2} \\times 3\\sqrt{6}$

**c)** $7\\sqrt{3} \\times 2\\sqrt{3}$

**d)** $6\\sqrt{5} \\times 3\\sqrt{10}$

**e)** $4\\sqrt{7} \\times 2\\sqrt{14}$

**f)** $10\\sqrt{2} \\times 3\\sqrt{8}$

**g)** $3\\sqrt{6} \\times 5\\sqrt{15}$

**h)** $8\\sqrt{3} \\times \\sqrt{12}$

**i)** $2\\sqrt{11} \\times 5\\sqrt{22}$

**j)** $9\\sqrt{7} \\times 2\\sqrt{21}$

Show Answers 

**a)** $8\\sqrt{15}$ **b)** $30\\sqrt{3}$ **c)** $42$ **d)** $90\\sqrt{2}$ **e)** $56\\sqrt{2}$

**f)** $120$ **g)** $45\\sqrt{10}$ **h)** $48$ **i)** $110\\sqrt{2}$ **j)** $126\\sqrt{3}$

*\[Video: "Surds (Part 2) Adding & Subtracting" goes here\]*

You can only add or subtract surds once they share the same number under the root — simplify first, then combine.

**Worked Example**

$\\sqrt{12} + \\sqrt{27} = 2\\sqrt{3} + 3\\sqrt{3} = 5\\sqrt{3}$

## Question 6

**a)** $\\sqrt{8} + \\sqrt{18}$

**b)** $\\sqrt{50} + \\sqrt{8}$

**c)** $\\sqrt{75} + \\sqrt{27}$

**d)** $\\sqrt{200} - \\sqrt{32}$

**e)** $\\sqrt{8} + \\sqrt{2} + \\sqrt{72}$

**f)** $\\sqrt{300} - \\sqrt{48}$

**g)** $\\sqrt{1000} + \\sqrt{90}$

**h)** $\\sqrt{28} + \\sqrt{63}$

**i)** $\\sqrt{20} + \\sqrt{45}$

**j)** $\\sqrt{54} - \\sqrt{24}$

Show Answers 

**a)** $5\\sqrt{2}$ **b)** $7\\sqrt{2}$ **c)** $8\\sqrt{3}$ **d)** $6\\sqrt{2}$ **e)** $9\\sqrt{2}$

**f)** $6\\sqrt{3}$ **g)** $13\\sqrt{10}$ **h)** $5\\sqrt{7}$ **i)** $5\\sqrt{5}$ **j)** $\\sqrt{6}$

## Question 7

Simplify each of the following, using coefficients in front of the surds.

**a)** $3\\sqrt{8} + \\sqrt{2}$

**b)** $4\\sqrt{27} - \\sqrt{75}$

**c)** $2\\sqrt{50} + 5\\sqrt{32}$

**d)** $\\sqrt{200} - 3\\sqrt{18}$

**e)** $4\\sqrt{80} + 3\\sqrt{45}$

**f)** $6\\sqrt{75} - 2\\sqrt{12}$

**g)** $10\\sqrt{7} + 2\\sqrt{175}$

**h)** $3\\sqrt{28} + 2\\sqrt{63}$

**i)** $5\\sqrt{48} - 2\\sqrt{27}$

**j)** $4\\sqrt{54} - 3\\sqrt{24}$

Show Answers 

**a)** $7\\sqrt{2}$ **b)** $7\\sqrt{3}$ **c)** $30\\sqrt{2}$ **d)** $\\sqrt{2}$ **e)** $25\\sqrt{5}$

**f)** $26\\sqrt{3}$ **g)** $20\\sqrt{7}$ **h)** $12\\sqrt{7}$ **i)** $14\\sqrt{3}$ **j)** $6\\sqrt{6}$

*\[Video: "Surds (Part 4) Expanding Double Brackets" goes here\]*

Expand brackets containing surds the same way as with algebra — multiply every term in the first bracket by every term in the second, then simplify.

**Worked Example**

$(2+\\sqrt{3})(1-\\sqrt{3}) = 2 - 2\\sqrt{3} + \\sqrt{3} - 3 = -1 - \\sqrt{3}$

## Question 8

**a)** $(3+\\sqrt{5})(4-\\sqrt{5})$

**b)** $(\\sqrt{7}-1)(\\sqrt{7}-1)$

**c)** $(5-\\sqrt{3})(5+\\sqrt{3})$

**d)** $(\\sqrt{7}+\\sqrt{2})(\\sqrt{8}+\\sqrt{7})$

**e)** $(1+2\\sqrt{2})(2-\\sqrt{2})$

**f)** $(3\\sqrt{5}+7)(2\\sqrt{5}+1)$

**g)** $(3+\\sqrt{2})^2$

**h)** $(1+\\sqrt{5})^2$

**i)** $(10-\\sqrt{2})^2$

**j)** $(2\\sqrt{3}-3)^2$

Show Answers 

**a)** $7+\\sqrt{5}$ **b)** $8-2\\sqrt{7}$ **c)** $22$ **d)** $11+3\\sqrt{14}$ **e)** $3\\sqrt{2}-2$

**f)** $37+17\\sqrt{5}$ **g)** $11+6\\sqrt{2}$ **h)** $6+2\\sqrt{5}$ **i)** $102-20\\sqrt{2}$ **j)** $21-12\\sqrt{3}$

*\[Video: "Surds (Part 5) Rationalising the Denominator 1" goes here\]*

To rationalise a single-term surd denominator, multiply top and bottom by that surd.

**Worked Example**

$\\dfrac{6}{\\sqrt{2}} = \\dfrac{6}{\\sqrt{2}} \\times \\dfrac{\\sqrt{2}}{\\sqrt{2}} = \\dfrac{6\\sqrt{2}}{2} = 3\\sqrt{2}$

## Question 9

**a)** $\\dfrac{1}{\\sqrt{5}}$

**b)** $\\dfrac{4}{\\sqrt{3}}$

**c)** $\\dfrac{12}{\\sqrt{6}}$

**d)** $\\dfrac{\\sqrt{2}}{\\sqrt{7}}$

**e)** $\\dfrac{3\\sqrt{5}}{\\sqrt{2}}$

**f)** $\\dfrac{8}{3\\sqrt{2}}$

**g)** $\\dfrac{10}{\\sqrt{8}}$

**h)** $\\dfrac{15}{\\sqrt{12}}$

**i)** $\\dfrac{20}{\\sqrt{50}}$

**j)** $\\dfrac{2\\sqrt{3}}{\\sqrt{27}}$

Show Answers 

**a)** $\\dfrac{\\sqrt{5}}{5}$ **b)** $\\dfrac{4\\sqrt{3}}{3}$ **c)** $2\\sqrt{6}$ **d)** $\\dfrac{\\sqrt{14}}{7}$ **e)** $\\dfrac{3\\sqrt{10}}{2}$

**f)** $\\dfrac{4\\sqrt{2}}{3}$ **g)** $\\dfrac{5\\sqrt{2}}{2}$ **h)** $\\dfrac{5\\sqrt{3}}{2}$ **i)** $2\\sqrt{2}$ **j)** $\\dfrac{2}{3}$

*\[Video: "Surds (Part 6) Rationalising the Denominator 2" goes here\]*

To rationalise a two-term denominator, multiply top and bottom by its conjugate (same terms, opposite sign in the middle).

**Worked Example**

$\\dfrac{2}{3+\\sqrt{2}} = \\dfrac{2}{3+\\sqrt{2}} \\times \\dfrac{3-\\sqrt{2}}{3-\\sqrt{2}} = \\dfrac{2(3-\\sqrt{2})}{7} = \\dfrac{6-2\\sqrt{2}}{7}$

## Question 10

**a)** $\\dfrac{1}{3+\\sqrt{2}}$

**b)** $\\dfrac{6}{4-\\sqrt{5}}$

**c)** $\\dfrac{10}{\\sqrt{7}-2}$

**d)** $\\dfrac{12}{\\sqrt{3}+1}$

**e)** $\\dfrac{\\sqrt{5}}{6+\\sqrt{5}}$

**f)** $\\dfrac{2\\sqrt{3}}{\\sqrt{3}-1}$

**g)** $\\dfrac{3+\\sqrt{2}}{5-\\sqrt{2}}$

**h)** $\\dfrac{\\sqrt{3}+1}{\\sqrt{3}-1}$

**i)** $\\dfrac{1}{\\sqrt{7}-\\sqrt{3}}$

**j)** $\\dfrac{4\\sqrt{2}}{\\sqrt{5}+\\sqrt{3}}$

Show Answers 

**a)** $\\dfrac{3-\\sqrt{2}}{7}$ **b)** $\\dfrac{24+6\\sqrt{5}}{11}$ **c)** $\\dfrac{20+10\\sqrt{7}}{3}$ **d)** $6\\sqrt{3}-6$ **e)** $\\dfrac{6\\sqrt{5}-5}{31}$

**f)** $3+\\sqrt{3}$ **g)** $\\dfrac{17+8\\sqrt{2}}{23}$ **h)** $2+\\sqrt{3}$ **i)** $\\dfrac{\\sqrt{7}+\\sqrt{3}}{4}$ **j)** $2\\sqrt{10}-2\\sqrt{6}$

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