Roots as Inverse Growth Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
Between which two consecutive integers does \(\sqrt{75}\) lie? Estimate to one decimal.

Solution

  1. 1
    \(8^2 = 64 < 75 < 81 = 9^2\), so \(8 < \sqrt{75} < 9\).
  2. 2
    75 is \(\frac{75-64}{81-64} = \frac{11}{17} \approx 0.65\) of the way from 64 to 81.
  3. 3
    Estimate: \(8 + 0.65 \approx 8.7\). (Actual: \(\sqrt{75} \approx 8.66\))

Answer

Between 8 and 9; approximately 8.7
Bracketing: \(8^2=64 < 75 < 81=9^2\). Linear interpolation gives \(\approx 8.7\); exact value is \(\approx 8.66\).

About Roots as Inverse Growth

Roots reverse the process of exponentiation: the nnth root of aa finds the number that, raised to the nnth power, produces aa. For example, 83=2\sqrt[3]{8} = 2 because 23=82^3 = 8.

Learn more about Roots as Inverse Growth →

More Roots as Inverse Growth Examples

Example 1 easy

Since (8^2 = 64), what is (sqrt{64})? Explain roots as inverses of powers.

Example 2 medium

Estimate (sqrt{50}) to one decimal place by finding the two perfect squares it lies between.

Example 3 easy

Find (sqrt{144}) and (sqrt[3]{27}).

← All Roots as Inverse Growth Examples Roots as Inverse Growth Concept