Question

Put these numbers in order from least to greatest.
3(the square root sign)155 7.24(24 repeated) (the square root sign)23

Answer 1

To order the numbers \(3\sqrt{155}\), \(7.24\overline{24}\), and \(\sqrt{23}\) from least to greatest, we can first evaluate each expression.

1. **Calculating \(3\sqrt{155}\)**:

\[
\sqrt{155} \approx 12.449
\]
\[
3\sqrt{155} \approx 3 \times 12.449 \approx 37.347
\]

2. **Calculating \(7.24\overline{24}\)**:

The repeating decimal \(7.24\overline{24}\) can be expressed as:
- Let \(x = 7.242424...\)
- Multiply by 100 to shift the decimal point: \(100x = 724.242424...\)
- Now, subtract \(x\) from \(100x\):
\[
100x - x = 724.242424... - 7.242424...
\]
\[
99x = 717
\]
\[
x = \frac{717}{99} \approx 7.24242
\]

3. **Calculating \(\sqrt{23}\)**:

\[
\sqrt{23} \approx 4.795
\]

Now, we can compare the evaluated values:
- \( \sqrt{23} \approx 4.795 \)
- \( 7.24\overline{24} \approx 7.24242 \)
- \( 3\sqrt{155} \approx 37.347 \)

Putting them in order from least to greatest, we have:

\[
\sqrt{23} < 7.24\overline{24} < 3\sqrt{155}
\]

Thus, the final ordered list is:

\[
\sqrt{23}, \quad 7.24\overline{24}, \quad 3\sqrt{155}
\]