3.2 Estimating powers and roots — Learn

Prerequisites: square and cube numbers, place value and rounding.

Objectives: estimate powers and roots; give a reason for an estimate by comparing nearby values.

By the end of this section, you can estimate powers and roots and explain why your estimate is reasonable.

This section covers N6.

Compare with known values

To estimate a root, find two nearby powers you know. The root lies between the matching base values. You can then refine the estimate.

Example 1. Estimate $\sqrt{70}$ to the nearest whole number.

So $\sqrt{70}$ lies between 8 and 9. The midpoint of 64 and 81 is 72.5. Since $70<72.5$, $\sqrt{70}$ rounds to 8.

Example 2. Estimate $\sqrt[3]{200}$ to the nearest whole number.

So $\sqrt[3]{200}$ lies between 5 and 6. The midpoint of 125 and 216 is 170.5. Since $200>170.5$, $\sqrt[3]{200}$ rounds to 6.

Estimating powers

You can estimate a power by rounding its base first. State the rounded base and then calculate its power.

Example 3. Estimate $19^3$ by rounding 19 to one significant figure.

The estimate is 8000. The exact value is $19^3=6859$, so rounding the base gives an estimate that is 1141 too high.

Watch out. A nearby number can give a poor estimate when the power grows quickly. Rounding 19 to 20 changes the cube from $6859$ to $8000$. State the estimate method and check that its accuracy suits the question.

Progress check

After 20 doublings, the file size is $2^{20}$. Since $2^{10}=1024$, predict whether it is closer to $10^3$ or $10^6$ units. The next section gives a way to rewrite the power.

You should now be able to

• locate a square or cube root between consecutive whole numbers;
• round a root to the nearest whole number by comparing with the midpoint of nearby powers;
• estimate a power by rounding its base;
• explain why an estimate is reasonable.