Question

Estimate √50 to the hundredths place.

1. Estimate between two whole numbers: 72 = 49, 82 = 64
2. Estimate further to the tenths place: 7.02 = 49.0, 7.12 = 50.41
3. Estimate further to the hundredths place:
The √50 is between
and
.

Answer 1

To estimate \( \sqrt{50} \) to the hundredths place, let's follow the steps you've outlined:

1. **Estimate between two whole numbers**:
Since \( 7^2 = 49 \) and \( 8^2 = 64 \), we know that \( \sqrt{50} \) falls between \( 7 \) and \( 8 \).

2. **Estimate further to the tenths place**:
We take decimals between 7 and 8 to narrow it down. Specifically:
- For \( 7.0 \):
\[
7.0^2 = 49.00
\]
- For \( 7.1 \):
\[
7.1^2 = 50.41 \quad (\text{too high})
\]
Since \( \sqrt{50} \) is between \( 7.0 \) and \( 7.1 \), we confirm our range is now between \( 7.0 \) and \( 7.1 \).

3. **Estimate further to the hundredths place**:
We can dig deeper by checking values like \( 7.05 \) and \( 7.06 \):
- For \( 7.05 \):
\[
7.05^2 = 49.7025 \quad (\text{still less than } 50)
\]
- For \( 7.06 \):
\[
7.06^2 = 49.8436 \quad (\text{still less than } 50)
\]
- For \( 7.07 \):
\[
7.07^2 = 49.9849 \quad (\text{still less than } 50)
\]
- For \( 7.08 \):
\[
7.08^2 = 50.0064 \quad (\text{too high})
\]

Since \( 7.07^2 < 50 < 7.08^2 \), the square root of 50 is between \( 7.07 \) and \( 7.08 \).

Therefore, the final statement would be:

The \( \sqrt{50} \) is between **7.07** and **7.08**.