Powers and roots Quiz (Grade 1)

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Topic:

Score: 0 / 10

Powers and roots — Notes & Worked Examples Show / Hide

Explanation:

A square number (n²) is the result of multiplying an integer by itself (first ten: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100). A cube number (n³) comes from multiplying an integer by itself three times: n × n × n (first five: 1, 8, 27, 64, 125). Never multiply the base by the power (4³ ≠ 12).

Roots are the inverse of powers: the square root (√x) asks what positive number squared gives x, and the cube root (∛x) asks what number cubed gives x. Also remember that any non-zero number to the power of 0 equals 1 (x⁰ = 1).

Example 1: Work out the value of 7² + √81 − 2³

Solution:

Evaluate each power or root separately first:

• 7² = 7 × 7 = 49

• √81 = 9 (since 9 × 9 = 81)

• 2³ = 2 × 2 × 2 = 8

Combine: 49 + 9 − 8 = Answer: 50

Example 2: Work out the value of 4³ − ∛64 + 10³

Solution:

Evaluate each term carefully:

• 4³ = 4 × 4 × 4 = 64

• ∛64 = 4 (since 4 × 4 × 4 = 64)

• 10³ = 10 × 10 × 10 = 1000

Combine: 64 − 4 + 1000 = Answer: 1060

One Page Guide (PDF) Powers and roots Worksheets

Question 1

[1 mark]

Jason says,
"The cube root of 27 is 9 because 27 divided by 3 is 9"

Is Jason correct?
You must give a reason for your answer.

Worked Solution & Answer

No, because 3 × 3 × 3 = 27 so the cube root of 27 is 3. Finding a cube root is not dividing by 3. B1

Answer: No with reason (cube root of 27 is 3, not 9)

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Question 2

[1 mark]

Find the square root of 64

Worked Solution & Answer

8 × 8 = 64 ⇒ √64 = 8 B1

Answer: 8

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Question 3 OCR Nov 2024 3F Q4

[2 marks]

Ben thinks of a number.
Ben says,
“When I square root my number and divide the result by 10 the answer is 1.3.”
Find Ben’s number.

Worked Solution & Answer

1.3 × 10 = 13 M1
13² = 169. A1

Answer: 169

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Question 4 AQA Nov 2023 3F Q22 / 3H Q7

[3 marks]

The first two cube numbers are 1 and 8

Show that
the 3rd cube number can be written as the sum of three different prime numbers.

Worked Solution & Answer

3rd cube number = 3³ = 27. M2
27 = 3 + 5 + 19 (or 3 + 7 + 17 or 3 + 11 + 13 or 5 + 11 + 11 — must be different primes: 3 + 5 + 19, 3 + 7 + 17, or 3 + 11 + 13). C1

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Question 5 AQA Jun 2024 2F Q12

[3 marks]

A code has five different digits written in order, starting with the smallest.
The last digit is the only square number.
The middle digit is the only even number.

Work out the code.

Answer

Worked Solution & Answer

The middle digit is the only even number, so the other 4 digits are odd M1
Odd digits are 1, 3, 5, 7, 9. The only square number among them is 1 and 9.
Since the last digit is the only square number, 1 cannot be used
Thus the odd digits must be 3, 5, 7, 9. M1
So the digits in order are 3, 5, [even], 7, 9.
The even digit between 5 and 7 must be 6.
Code: 3 5 6 7 9. A1

Answer: Answer 3 5 6 7 9

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Question 6

[4 marks]

The diagram shows a shape made from 6 identical squares.

The total area of the shape is 5406 cm²

(a) Find an estimate for the length of one side of each square.
Give your answer correct to the nearest whole number. [3 marks]

(b) Is your answer to part (a) an underestimate or an overestimate?
You must give a reason for your answer. [1 mark]

Worked Solution & Answer

(a)

5406 ÷ 6 = 901 ≈ 900 M2
√900 = 30 cm A1

Answer (a): 30 cm

(b)

√900 < √901 so 30 is an underestimate C1

Answer (b): Underestimate because 5406 was rounded down to 5400 (or 901 was rounded down to 900)

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Question 7 AQA Jun 2024 1H Q15

[4 marks]

(a) Between which two consecutive integers does √210 lie? [1 mark]
15 (b) Use approximations to show that 1.92⁷ + 6.9³ < 5 × ∛1 000 350. [3 marks]

(a) Between which two **consecutive** integers does the square root of 210 lie? [1 mark]

(b) Here are two calculations, A and B.
@@TABULAR₀@@
Use approximations to show that answer to A < answer to B [3 marks]

Worked Solution & Answer

(a)

14² = 196 and 15² = 225, so √210 lies between 14 and 15 B1

Answer (a): 14 and 15

(b)

A < 2⁷ + 7³ = 128 + 343 = 471 M1
B > 5 × ∛1 000 000 = 5 × 100 = 500 M1
Since 471 < 500, answer to A < answer to B C1

Answer (b): A ≈ 2⁷ + 7³ = 128 + 343 = 471, B ≈ 5 × 100 = 500, and 471 < 500

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Question 8 AQA Nov 2020 2F Q8

[4 marks]

Work out
(a) Write down an even whole number that is also a square number.
8 (b) Write down all the cube numbers between 100 and 400
8 (c) Write down two numbers that are multiples of 3 and multiply to make 216

(a) Write down an even whole number that is also a square number. [1 mark]

Answer

(b) Write down all the cube numbers between 100 and 400 [2 marks]

Answer

Worked Solution & Answer

(a)

4 is an even number and 2² = 4 (or 16, 36, 64). B1

Answer (a): Answer 4

(b)

4³ = 64 M1
5³ = 125
6³ = 216
7³ = 343
8³ = 512
Between 100 and 400: 125, 216, 343. A1

Answer (b): Answer 125, 216, 343

(c)

Multiples of 3 with product 216: e.g. 6 × 36 = 216 (or 12 and 18, 9 and 24, 3 and 72). B1

Answer (c): Answer 6 and 36

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Question 9 AQA Jun 2025 1F Q2

[4 marks]

Here is a card from a game.

(a) Write down the number from the card that is a multiple of 5 [1 mark]

Answer

(b) Write down the number from the card that is a factor of 40 [1 mark]

Answer

(d) Write down the number from the card that is a square number. [1 mark]

Answer

Worked Solution & Answer

(a)

15 ends in 5, so multiple of 5 B1

Answer (a): Answer 15

(b)

40 ÷ 8 = 5, so 8 is a factor of 40 B1

Answer (b): Answer 8

(c)

13 has only 1 and 13 as factors, so prime B1

Answer (c): Answer 13

(d)

6² = 36, so 36 is a square number B1

Answer (d): Answer 36

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Question 10 IGCSE Jan 2021 2F Q1

[5 marks]

Here is a list of numbers in a box.

(a) a multiple of 11 [1 mark]

(b) a factor of 30 [1 mark]

(d) a prime number [1 mark]

(e) two numbers whose sum is 84 [1 mark]

Worked Solution & Answer

(a)

44 = 4 × 11 B1

Answer (a): 44

(b)

30 ÷ 6 = 5 B1

Answer (b): 6

(c)

36 = 6² B1

Answer (c): 36

(d)

17 is prime (91 = 7 × 13) B1

Answer (d): 17

(e)

8 + 76 = 84 B1

Answer (e): 8 and 76

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Score: 0 / 10 (0 / 31 marks)

Powers and roots Worksheets Next: BIDMAS →

Grade 1 Quizzes

16 Topics

1A - Addition	1B - Subtraction
1C - Multiplication	1D - Division
1E - Time	1F - Ordering FDP
1G - Place Value	1H - Rounding
1J - Negative numbers	1K - Powers and roots
1L - BIDMAS	1M - Factors, multiples and primes
1N - Coordinates	1P - Pictograms
1Q - 2D shapes	1R - 3D shapes