Indices and Surds — Solutions

Full worked solutions, with method and final answer for every part.

Question 1

(a)A negative power means the reciprocal: 2⁻³ = 1/2³.

1/8

(b)The power 2/3 means: cube root first, then square. ∛8 = 2, then 2² = 4.

Question 2

(a)Find the largest square factor of 50: 50 = 25 × 2, and √25 = 5.

√50 = 5√2

Question 3

(a)Simplify each surd separately: √12 = √(4×3) = 2√3, and √27 = √(9×3) = 3√3, then add.

2√3 + 3√3 = 5√3

Question 4

(a)3⁻² = 1/3².

1/9

Question 5

(a)A power of −1/2 means take the reciprocal, then the square root: reciprocal of 1/4 is 4, √4=2.

Question 6

(a)Find the largest square factor: 75 = 25 × 3.

√75 = 5√3

Question 7

(a)Multiply under one root: √8 × √2 = √16.

√16 = 4

Question 8

(a)Multiply top and bottom by √2.

(1×√2)/(√2×√2) = √2/2

Question 9

(a)Square the coefficient and the surd separately: 5²=25, (√3)²=3.

25 × 3 = 75

Question 10

(a)Simplify each surd: √45 = 3√5, √20 = 2√5, then subtract.

3√5 − 2√5 = √5

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