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Fractional & negative indices practice questions by difficulty

Question bank · Higher · Grade 6 · Non-calculator

Fractional & negative indices: questions only, sorted from first steps to stretch. Count: 6 questions (2 warm-up, 2 core, 2 stretch, 1 to 2 marks each), chosen so they do not repeat the topic page. Also here: a multi-part show-that question, 15 quick checks. Pair it with the Fractional & negative indices topic page and the Fractional & negative indices worked examples.

How to use this page: Try the stretch set first; if you drop marks, come back down to core and find which step went wrong.

Warm-up questions

2 questions, 1 mark each. The first one: “Work out 25(1/2).”

Question 1 Easy1 mark

Work out 251/2.

Show answer

Answer: 5

  1. A power of 1/2 means square root.
  2. Square root of 25 is 5.
  • Answer (1): Answer 5

Question 2 Easy1 mark

Work out 81/3.

Show answer

Answer: 2

  1. A power of 1/3 means cube root.
  2. 81/3 = 2 because 2 cubed is 8.
  • Answer (1): Answer 2

Core questions

2 questions, 2 marks each. Starting with: “Work out 5(-2).”

Question 3 Medium2 marks

Work out 5-2.

Show answer

Answer: 1/25

  1. Calculate 52 = 25.
  2. Apply reciprocal: 1/25.
  • Method (1): Calculate 52 = 25
  • Accuracy (1): Answer 1/25

Question 4 Medium2 marks

Work out 2-3.

Show answer

Answer: 1/8

  1. Calculate 23 = 8.
  2. Apply reciprocal: 1/8.
  • Method (1): Calculate 23 = 8
  • Accuracy (1): Answer 1/8

Stretch questions

2 questions, 2 marks each. It opens with: “Work out 64(-2/3).”

Question 5 Hard2 marks

Work out 64-2/3.

Show answer

Answer: 1/16

  1. Cube root of 64 is 4, then square to get 16.
  2. Apply negative power: 1/16.
  • Method (1): Cube root of 64 is 4, then square to get 16
  • Accuracy (1): Answer 1/16

Question 6 Hard2 marks

Work out 81-1/2.

Show answer

Answer: 1/9

  1. Square root of 81 is 9.
  2. Apply negative power: 1/9.
  • Method (1): Square root of 81 is 9
  • Accuracy (1): Answer 1/9

Multi-part exam-style question

2-part question for 4 marks; a later part may use an earlier answer. An exam-style question written by Yumio.

Question 7 Exam-style · Show that4 marks

Show all your working. Part (b) prints the answer; you earn the marks by reaching it.

(a) Work out 52. [1 mark]

(b) Use your answer to part (a). Work out 125-2/3. Show that the answer is 1/25. [3 marks]

Show answer and worked solution

Part (a)

Answer: 25

  1. Square the root: 52 = 25.

Marks for this part:

  • B1: 25

Part (b)

Answer: Shown: the working reaches 1/25

  1. Take the cube root of 125: cube root of 125 = 5.
  2. The negative index means one over that: 1/25.

Marks for this part:

  • M1: Take the cube root of 125: cube root of 125 = 5
  • M1: The negative index means one over that: 1/25
  • A1: reaches 1/25 with every line correct

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

What does the bottom of a fractional index tell you?

Show answer

It is the root you take first. It is the root you take first.

Check 2

Cube root then square: what does 82/3 equal?

Show answer

Cube root of 8 is 2, squared is 4. Cube root of 8 is 2, squared is 4.

Check 3

Fourth root then cube: what does 163/4 equal?

Show answer

Fourth root of 16 is 2, cubed is 8. Fourth root of 16 is 2, cubed is 8.

Check 4

Square-root power: 91/2 equals

  1. 16-1/2 = 1/4
  2. 81/3 = 2
  3. 91/2 = 3
Show answer

91/2 = 3 Square root of 9 is 3.

Check 5

Cube-root power: 81/3 equals

  1. 81-1/2 = 1/9
  2. 81/3 = 2
  3. 27-2/3 = 1/9
Show answer

81/3 = 2 Cube root of 8 gives 2.

Check 6

Five factors of two: 25 equals

Show answer

2 × 2 × 2 × 2 × 2 = 32. 2 × 2 × 2 × 2 × 2 = 32.

Check 7

Four factors of three: 34 equals

Show answer

3 × 3 × 3 × 3 = 81. 3 × 3 × 3 × 3 = 81.

Check 8

What does a negative index do to the value?

Show answer

It takes the reciprocal of the positive power. It takes the reciprocal of the positive power.

Check 9

Reciprocal of a square: 4-2 equals

Show answer

4-2 = 1 / 42 = 1/16. 4-2 = 1 / 42 = 1/16.

Check 10

Reciprocal of a cube: 2-3 equals

Show answer

2-3 = 1 / 23 = 1/8. 2-3 = 1 / 23 = 1/8.

Check 11

Negative fractional power 27-2/3 equals

  1. 81-1/2 = 1/9
  2. 27-2/3 = 1/9
  3. 8-2/3 = 1/4
Show answer

27-2/3 = 1/9 27-2/3 = 1 / 32 = 1/9.

Check 12

Negative fractional power 8-2/3 equals

  1. 81/3 = 2
  2. 8-2/3 = 1/4
  3. 81-1/2 = 1/9
Show answer

8-2/3 = 1/4 8-2/3 = 1 / 22 = 1/4.

Check 13

Negative square-root power 16-1/2 equals

  1. 16-1/2 = 1/4
  2. 27-2/3 = 1/9
  3. 91/2 = 3
Show answer

16-1/2 = 1/4 16-1/2 = 1 / √16 = 1/4.

Check 14

Negative square-root power 81-1/2 equals

  1. 81/3 = 2
  2. 8-2/3 = 1/4
  3. 81-1/2 = 1/9
Show answer

81-1/2 = 1/9 81-1/2 = 1 / √81 = 1/9.

Check 15

Why is 4-2 not equal to -16?

Show answer

The minus is on the index, so the result is a fraction. The minus is on the index, so the result is a fraction.

Keep going with Fractional & negative indices

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