To order the numbers \(3\sqrt{155}\), \(7.24\overline{24}\), and \(\sqrt{23}\) from least to greatest, we can first evaluate each expression.
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Calculating \(3\sqrt{155}\):
\[ \sqrt{155} \approx 12.449 \] \[ 3\sqrt{155} \approx 3 \times 12.449 \approx 37.347 \]
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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 \(100x\): \[ 100x - x = 724.242424... - 7.242424... \] \[ 99x = 717 \] \[ x = \frac{717}{99} \approx 7.24242 \]
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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:
\[ \sqrt{23}, \quad 7.24\overline{24}, \quad 3\sqrt{155} \]