Which of the following pairs of numbers provide the best estimate of square root 53?

Answer 1

To estimate the square root of 53, let's find the closest perfect squares that are near 53.

We know that:

\( 7^2 = 49 \)
\( 8^2 = 64 \)

Since 49 is less than 53 and 64 is greater than 53, we know that the square root of 53 must be somewhere between 7 and 8.

Now we can use interpolation to find a better estimate. The difference between 49 and 64 is 15, and the difference between 53 and 49 is 4. To find how much more than 7 the square root of 53 is, we can use the formula:

\( \sqrt{53} \approx 7 + \frac{4}{15} \times (8 - 7) \)

This gives us:

\( \sqrt{53} \approx 7 + \frac{4}{15} \)

Calculating the fraction:

\( \frac{4}{15} \approx 0.2667 \)

Adding that to 7, we get:

\( \sqrt{53} \approx 7.2667 \)

Hence, a good estimate for the square root of 53 is 7.2667, which is between 7 and 8. If you're looking for the square root in terms of a simpler rational number without a calculator, an approximation slightly greater than 7.25 but less than 7.3 would be reasonable.