GCSE Maths | Surds

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

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

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

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

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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) $27\sqrt{3}$

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

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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]

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) $\sqrt{98}$

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

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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]

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

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

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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]

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

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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]

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

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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]

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

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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) $2\sqrt{10}-2\sqrt{6}$