To simplify the expression \((9 \frac{2}{3})^{\frac{3}{4}}\), let's follow the steps:
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Convert the Mixed Number: First, convert \(9 \frac{2}{3}\) into an improper fraction: \[ 9 \frac{2}{3} = 9 + \frac{2}{3} = \frac{27}{3} + \frac{2}{3} = \frac{29}{3} \]
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Substitute Back into the Expression: Replace \(9 \frac{2}{3}\) with \(\frac{29}{3}\) in the expression: \[ (9 \frac{2}{3})^{\frac{3}{4}} = \left(\frac{29}{3}\right)^{\frac{3}{4}} \]
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Apply the Rational Exponent: When using rational exponents, we can rewrite the expression: \[ \left(\frac{29}{3}\right)^{\frac{3}{4}} = \frac{29^{\frac{3}{4}}}{3^{\frac{3}{4}}} \]
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Calculate the Exponents:
- \(29^{\frac{3}{4}}\) can be understood as the fourth root of \(29^3\).
- \(3^{\frac{3}{4}}\) can be understood as the fourth root of \(3^3\).
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Express the Results: \[ \frac{29^{\frac{3}{4}}}{3^{\frac{3}{4}}} = \frac{\sqrt[4]{29^3}}{\sqrt[4]{3^3}} = \frac{\sqrt[4]{29^3}}{\sqrt[4]{27}} = \frac{\sqrt[4]{29^3}}{3^{\frac{3}{4}}} \]
Thus, the simplified form is: \[ \left(\frac{29}{3}\right)^{\frac{3}{4}} = \frac{29^{\frac{3}{4}}}{3^{\frac{3}{4}}} \]
You can also leave it in this form or calculate approximate numerical values if needed, but as a simplification, this is a neat way to present it.