Estimate the cube root of the following numbers up to one decimal place. (i) 345 (ii) 1275 (iii) 8998
The Correct Answer and Explanation is:
Estimated Cube Roots (to one decimal place):
(i) ∛345 ≈ 7.0
(ii) ∛1275 ≈ 10.9
(iii) ∛8998 ≈ 20.8
Explanation:
To estimate the cube root of a number up to one decimal place, we identify two perfect cubes between which the number lies, then use trial and error or interpolation to narrow down the decimal.
(i) ∛345
We know:
- ∛216 = 6 (because 6³ = 216)
- ∛343 = 7 (because 7³ = 343)
- ∛512 = 8 (because 8³ = 512)
Since 345 is just above 343, its cube root will be a bit more than 7. Trying 7.0:
- 7.0³ = 343
Trying 7.1: - 7.1³ = 357.911
So, ∛345 is slightly more than 7.0 but less than 7.1. Since 345 is very close to 343, we round down: - ∛345 ≈ 7.0
(ii) ∛1275
We know:
- ∛1000 = 10 (10³ = 1000)
- ∛1331 = 11 (11³ = 1331)
So, 1275 lies between 1000 and 1331. Try 10.8: - 10.8³ = 1259.712
Try 10.9: - 10.9³ = 1295.029
1275 lies between these two values, but closer to 10.8.9, so we choose:
- ∛1275 ≈ 10.9
(iii) ∛8998
We know:
- ∛8000 = 20 (20³ = 8000)
- ∛9261 = 21 (21³ = 9261)
So, 8998 is between 8000 and 9261. Try 20.8: - 20.8³ = 8991.872
Try 20.9: - 20.9³ = 9125.129
Since 8998 is closer to 8991.872, we choose:
- ∛8998 ≈ 20.8
Summary:
Estimation uses perfect cubes as reference points and checks nearby decimals. We approximate by testing numbers just above or below the boundaries. This method helps when a calculator is not available and reinforces number sense.
