Powers and roots worksheet — GCSE Higher

1.Given that 2⁵ = 32, work out 2⁶.

2.4ˣ = 64. Work out the value of x.

3.Work out 5⁰ + 5¹ + 5²

4.Four students each estimate √70 without a calculator. Ben says: '8² = 64 and 9² = 81, and 70 is about a third of the way from 64 to 81, so √70 ≈ 8.3.' Chloe says: '8.5² = 72.25, which is close to 70 but a little too big, so I'll bring it down slightly to √70 ≈ 8.4.' Dan says: '9² = 81 is closer to 70 than 8² = 64 is, so √70 ≈ 8.9.' Ella says: '8² = 64 and 9² = 81, so I'll just take the midpoint of 8 and 9: √70 ≈ 8.5.' Given that √70 = 8.3666... to 4 decimal places, whose estimate is closest?

5.Work out 6² − 4².

6.Work out −(−3)⁴ + (−3)³

7.Work out ⁴√16

8.Without using a calculator, estimate √20 × √12, giving your answer to the nearest whole number.

9.Work out ∛(−27) + ∛8

10.Work out √225 ÷ 3.

11.Work out the exact value of √(2² + 3²)

12.Given that 5³ = 125 and 6³ = 216, use a midpoint test to estimate ∛130 to 1 decimal place.

13.Without using a calculator, estimate the value of √70 × ∛65, giving your answer to 1 significant figure.

14.Work out √(4 × 9)

15.By first working out 2.9², then squaring your result, estimate 2.9⁴ to 2 significant figures.