Question

To two decimal places, the square root of 78 must lie between ___ and ___ because

Answer 1

To find the square root of 78 to two decimal places, we first need to determine two perfect squares that 78 falls between:

1. The square of 8 is \( 8^2 = 64 \).
2. The square of 9 is \( 9^2 = 81 \).

Since \( 64 < 78 < 81 \), we can conclude that:
\[
8 < \sqrt{78} < 9
\]

Next, we can narrow it down further by estimating between 8 and 9. We can try out some decimal values:

- \( 8.7^2 = 75.69 \)
- \( 8.8^2 = 77.44 \)
- \( 8.9^2 = 79.21 \)

From these calculations, we see that:
\[
8.8 < \sqrt{78} < 8.9
\]

Next, we can look at values in the vicinity of 8.8:

- \( 8.85^2 = 78.2225 \)
- \( 8.84^2 = 78.0224 \)

From this, we find:
\[
8.84 < \sqrt{78} < 8.85
\]

Thus, rounding to two decimal places, we find:
\[
\sqrt{78} \approx 8.83
\]

So, we can confidently say that the square root of 78 must lie between **8.83 and 8.84** because \( 8.83^2 \) is less than 78 and \( 8.84^2 \) is greater than 78.