Estimating the power of a number involves rounding the number to one significant figure (1sf) and multiplying that number a suitable number of times.
Estimating the root of a number uses a technique known as Trial and Improvement. The calculation works by taking a guess at the answer. The guess of that answer is raised to that power and checked against the original number. The guess is revised, and the calculation repeated until sufficient accuracy is obtained.
Trial and Improvement is best calculated using a table.
Estimate 4.4223.
Round 4.422 to 1 significant figure 4 x 4 x 4 = 64. The actual answer to 4.4223 is 84.468 (to 3dp).
Answer: 64
Estimate the value of `root(4)366218` to 2 significant figures.
Guess 20 as a starting value
| x | x4 | Note | |
| 20 | 160 000 | Too low | |
| 30 | 810 000 | Too high | |
| 25 | 390 625 | Too high | |
| 24 | 331 776 | Too low (see below) | |
| 24.5 | 360 300 | Too low |
Note: the answer is either 24 or 25 (to 2 significant figures). Test the halfway point between 24 and 25. 24.5 is also too low, therefore the number must be higher than 24.5. The number must be between 24.5 and 25, and to two significant figures it must be 25.
Answer: 25
See also Estimating Answers and Approximate Solutions using Iteration