Power and Roots (Copy)
Practice Questions – 1.3 Powers and Roots
Question 1
Write down the value of 12².
Question 2
Evaluate √144.
Question 3
Find the value of 7³.
Question 4
Work out ∛125.
Question 5
Calculate 15² − 14².
Question 6
Simplify √196 ÷ √49.
Question 7
Write down the cube of 10 and the cube root of 1000.
Question 8
If a = 5, calculate a⁴.
Question 9
Without using a calculator, find the value of (13² + 12²).
Question 10
Find √(81 + 64).
Question 11
Evaluate 2⁵ × 2³.
Question 12
Simplify (3²)³.
Question 13
If x = 16, find √x + x½.
Question 14
Calculate (5³ ÷ 5²).
Question 15
Find the value of (4³ + 2⁵).
Question 16
Simplify √(49) + ∛(64).
Question 17
Work out (8¹/³)².
Question 18
Write down the square roots of 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
Question 19
Write down the cubes of 1, 2, 3, 4, 5, 10.
Question 20
If y = 9, calculate y³ − √y.
Question 21
Simplify (2⁴ × 3²) ÷ (2² × 3).
Question 22
Find the positive square root of 2.25.
Question 23
Evaluate (6² + 8²)½.
Question 24
Express 64 as a power of 2.
Question 25
If a = 27, show that ∛a = 3.
Answer key and explanations — 1.3 Powers and Roots
1.
12² = 144
Explanation: squaring means multiplying the number by itself → 12 × 12.
2.
√144 = 12
Explanation: square root is the inverse of squaring.
3.
7³ = 343
Explanation: cube = 7 × 7 × 7.
4.
∛125 = 5
Explanation: cube root of 125 is the number multiplied by itself three times to give 125.
5.
15² − 14² = 225 − 196 = 29
Explanation: difference of consecutive squares always equals their sum (29 here).
6.
√196 ÷ √49 = 14 ÷ 7 = 2
Explanation: simplify roots before dividing.
7.
Cube of 10 = 1000, cube root of 1000 = 10
Explanation: cube and cube root cancel each other.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
8.
If a = 5, a⁴ = 5 × 5 × 5 × 5 = 625
9.
13² + 12² = 169 + 144 = 313
10.
√(81 + 64) = √145 ≈ 12.04
Explanation: combine inside root before taking square root.
11.
2⁵ × 2³ = 2⁸ = 256
Explanation: same base multiplication → add powers.
12.
(3²)³ = 3⁶ = 729
Explanation: power of a power → multiply powers (2 × 3 = 6).
13.
x = 16, √x + x½ = 4 + 4 = 8
Explanation: both √x and x½ mean the same.
14.
5³ ÷ 5² = 5¹ = 5
Explanation: division with same base → subtract powers.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
15.
4³ + 2⁵ = 64 + 32 = 96
16.
√49 + ∛64 = 7 + 4 = 11
17.
(8¹/³)² = (∛8)² = 2² = 4
18.
Square roots:
√1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5, √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10, √121 = 11, √144 = 12, √169 = 13, √196 = 14, √225 = 15
Explanation: learn by heart from 1 to 15.
19.
Cubes:
1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 10³ = 1000
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
20.
y = 9 → y³ − √y = 729 − 3 = 726
21.
(2⁴ × 3²) ÷ (2² × 3) = 2² × 3¹ = 4 × 3 = 12
Explanation: cancel common powers when dividing.
22.
Positive square root of 2.25 = 1.5
Explanation: 1.5 × 1.5 = 2.25.
23.
(6² + 8²)½ = (36 + 64)½ = (100)½ = 10
Explanation: this is a Pythagoras triple.
24.
64 = 2⁶
Explanation: divide repeatedly by 2: 64 → 32 → 16 → 8 → 4 → 2 → 1 (6 divisions).
25.
a = 27, ∛a = ∛27 = 3
Explanation: 3³ = 27, so cube root is 3.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course