Work out the value of 25^(1)/(2).

Recall the unit fraction index rule: a^(1)/(2) = √(a) Apply the rule: 25^(1)/(2) = √(25) Evaluate the root: √(25) = 5 Answer: 5

Work out the value of 27^(1)/(3).

Recall the unit fraction index rule: a^(1)/(3) = [3]a Apply the rule: 27^(1)/(3) = [3]27 Evaluate the root: [3]27 = 3 Answer: 3

Work out the value of 16^(1)/(4).

Recall the unit fraction index rule: a^(1)/(4) = [4]a Apply the rule: 16^(1)/(4) = [4]16 Evaluate the root: 2 × 2 × 2 × 2 = 16 [4]16 = 2 Answer: 2

Write √(7) as a power of 7.

Recall the root-to-power rule: √(a) = a^(1)/(2) Apply the rule to 7: √(7) = 7^(1)/(2) State the answer: 7^(1)/(2) Answer: 7^(1)/(2)

Work out the value of 8^(2)/(3).

Recall the non-unit fraction rule: a^(m)/(n) = ([n]a)^m Take the cube root first: 8^(2)/(3) = ([3]8)^2 = 2^2 Square the result: 2^2 = 4 Answer: 4

Fractional indices Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Fractional indices questions. See exactly how to solve problems on unit fractional index, square roots, cube roots, fourth roots.

unit fractional indexsquare rootscube rootsfourth rootsroots as powersnon-unit fractional index

GCSE Higher70 questionsStep-by-step solutions

Question 1

1 markeasy

Work out the value of

25^{\frac{1}{2}}

Worked solution

Recall the unit fraction index rule
$a^{\frac{1}{2}} = \sqrt{a}$
An index of one half means the square root. This is the key fact for the whole topic.
Apply the rule
$25^{\frac{1}{2}} = \sqrt{25}$
So we need the square root of 25 — the number that multiplies by itself to give 25.
Evaluate the root
$\sqrt{25} = 5$
5 times 5 is 25, so the answer is 5.

Answer

5

Question 2

1 markeasy

Work out the value of

27^{\frac{1}{3}}

Worked solution

Recall the unit fraction index rule
$a^{\frac{1}{3}} = \sqrt[3]{a}$
An index of one third means the cube root — the number that multiplies by itself three times to give the base.
Apply the rule
$27^{\frac{1}{3}} = \sqrt[3]{27}$
So we need the cube root of 27.
Evaluate the root
$\sqrt[3]{27} = 3$
$3 \times 3 \times 3 = 27$ , so the answer is $3$ .

Answer

3

Question 3

1 markeasy

Work out the value of

16^{\frac{1}{4}}

Worked solution

Recall the unit fraction index rule
$a^{\frac{1}{4}} = \sqrt[4]{a}$
An index of one quarter means the fourth root — the number that multiplies by itself four times to give the base.
Apply the rule
$16^{\frac{1}{4}} = \sqrt[4]{16}$
So we need the fourth root of 16.
Evaluate the root
$2 \times 2 \times 2 \times 2 = 16 \ \Rightarrow\ \sqrt[4]{16} = 2$
Four 2s multiply to 16, so the answer is 2.

Answer

2

Question 4

1 markeasy

Write

\sqrt{7}

as a power of

7

Worked solution

Recall the root-to-power rule
$\sqrt{a} = a^{\frac{1}{2}}$
A square root can always be written as a power with index one half. The rule works both ways.
Apply the rule to 7
$\sqrt{7} = 7^{\frac{1}{2}}$
The base stays as 7 and the root becomes the index one half. It does not matter that 7 is not a square number.
State the answer
$7^{\frac{1}{2}}$
So the square root of 7 written as a power is 7 to the power one half.

Answer

7^{\frac{1}{2}}

Question 5

2 markseasy

Work out the value of

8^{\frac{2}{3}}

Worked solution

Recall the non-unit fraction rule
$a^{\frac{m}{n}} = (\sqrt[n]{a})^m$
The bottom of the fraction is the root and the top is the power. Doing the root first keeps the numbers small.
Take the cube root first
$8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2$
The cube root of $8$ is $2$ , because $2 \times 2 \times 2 = 8$ .
Square the result
$2^2 = 4$
Now apply the power on top: 2 squared is 4.

Answer

4

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