10. Evaluate: \( \begin{array}{llll}\text { (a) } 81^{\frac{1}{4}} & \text { (b) } 125^{\frac{2}{3}} & \text { (c) } 32^{-\frac{3}{5}} & \text { (d) }\left(\frac{32}{243}\right)^{-\frac{2}{5}}\end{array} \)
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Mind Expander
Let's evaluate each expression one at a time. (a) \( 81^{\frac{1}{4}} \) To find this, we can express 81 as \( 3^4 \): \[ 81^{\frac{1}{4}} = (3^4)^{\frac{1}{4}} = 3^{4 \cdot \frac{1}{4}} = 3^1 = 3 \] So, \( 81^{\frac{1}{4}} = 3 \). (b) \( 125^{\frac{2}{3}} \) We start by expressing 125 as \( 5^3 \): \[ 125^{\frac{2}{3}} = (5^3)^{\frac{2}{3}} = 5^{3 \cdot \frac{2}{3}} = 5^2 = 25 \] Thus, \( 125^{\frac{2}{3}} = 25 \). (c) \( 32^{-\frac{3}{5}} \) We know that 32 can be expressed as \( 2^5 \): \[ 32^{-\frac{3}{5}} = (2^5)^{-\frac{3}{5}} = 2^{5 \cdot -\frac{3}{5}} = 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \] Therefore, \( 32^{-\frac{3}{5}} = \frac{1}{8} \). (d) \( \left(\frac{32}{243}\right)^{-\frac{2}{5}} \) First, let’s rewrite the fraction: \[ \frac{32}{243} = \frac{2^5}{3^5} \] Now we evaluate it as follows: \[ \left(\frac{32}{243}\right)^{-\frac{2}{5}} = \left(\frac{2^5}{3^5}\right)^{-\frac{2}{5}} = \frac{3^{5 \cdot \frac{2}{5}}}{2^{5 \cdot \frac{2}{5}}} = \frac{3^2}{2^2} = \frac{9}{4} \] Thus, \( \left(\frac{32}{243}\right)^{-\frac{2}{5}} = \frac{9}{4} \). Summing it all up: - (a) \( 3 \) - (b) \( 25 \) - (c) \( \frac{1}{8} \) - (d) \( \frac{9}{4} \)