Powers and roots worksheet — GCSE Higher

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

2.Work out √225 ÷ 3.

3.4ˣ = 64. Work out the value of x.

4.The mass of a radioactive sample, in grams, n years after it was first weighed is modelled by M = 200 × (1/2)ⁿ. Work out the mass the model gives after 3 years.

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

6.By considering fourth powers, work out which two consecutive integers ⁴√200 lies between.

7.Work out ⁴√81

8.Work out 3³ + 2⁴.

9.√700 lies between which two consecutive integers?

10.Work out 5⁰ + 5¹ + 5²

11.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?

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

13.Given that 4³ = 64 and 5³ = 125, estimate ∛100 to 1 decimal place.

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

15.Given that 2⁵ = 32, work out 2⁶.