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