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10. Evaluate: \( \begin{array}{llll}\text { (a) } 81^{\frac{1}{4}} & \text { (b) } 125^{\frac{2}{3}} & \text { (c) } 32^{-\frac{3}{5}} & \text { (d) }\left(\frac{32}{243}\right)^{-\frac{2}{5}}\end{array} \)

Ask by Beck Johnson. in Curaçao
Jan 10,2025

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Answer

\( 3, 25, \frac{1}{8}, \frac{9}{4} \)

Solution

Let's evaluate each expression step-by-step: --- ### **(a) \( 81^{\frac{1}{4}} \)** **Solution:** The exponent \(\frac{1}{4}\) represents the fourth root of 81. 1. **Express 81 as a power of a smaller number:** \[ 81 = 3^4 \] 2. **Apply the exponent:** \[ 81^{\frac{1}{4}} = \left(3^4\right)^{\frac{1}{4}} = 3^{4 \times \frac{1}{4}} = 3^1 = 3 \] **Answer:** \( 3 \) --- ### **(b) \( 125^{\frac{2}{3}} \)** **Solution:** The exponent \(\frac{2}{3}\) indicates the cube root of 125, squared. 1. **Find the cube root of 125:** \[ 125 = 5^3 \Rightarrow \sqrt[3]{125} = 5 \] 2. **Square the result:** \[ \left(\sqrt[3]{125}\right)^2 = 5^2 = 25 \] **Answer:** \( 25 \) --- ### **(c) \( 32^{-\frac{3}{5}} \)** **Solution:** The negative exponent \(-\frac{3}{5}\) means we take the reciprocal of \(32^{\frac{3}{5}}\). 1. **Find \(32^{\frac{3}{5}}\):** - First, find the 5th root of 32. \[ 32 = 2^5 \Rightarrow \sqrt[5]{32} = 2 \] - Then, raise it to the 3rd power. \[ 2^3 = 8 \] 2. **Take the reciprocal due to the negative exponent:** \[ 32^{-\frac{3}{5}} = \frac{1}{32^{\frac{3}{5}}} = \frac{1}{8} \] **Answer:** \( \frac{1}{8} \) --- ### **(d) \( \left( \frac{32}{243} \right)^{-\frac{2}{5}} \)** **Solution:** The negative exponent \(-\frac{2}{5}\) indicates taking the reciprocal of \( \left( \frac{32}{243} \right)^{\frac{2}{5}} \). 1. **Express the fraction in terms of exponents:** \[ \frac{32}{243} = \frac{2^5}{3^5} \] 2. **Apply the exponent \(\frac{2}{5}\):** \[ \left( \frac{2^5}{3^5} \right)^{\frac{2}{5}} = \frac{2^{5 \times \frac{2}{5}}}{3^{5 \times \frac{2}{5}}} = \frac{2^2}{3^2} = \frac{4}{9} \] 3. **Take the reciprocal due to the negative exponent:** \[ \left( \frac{32}{243} \right)^{-\frac{2}{5}} = \frac{1}{\left( \frac{32}{243} \right)^{\frac{2}{5}}} = \frac{9}{4} \] **Answer:** \( \frac{9}{4} \) --- ### **Summary of Answers:** - **(a)** \( 3 \) - **(b)** \( 25 \) - **(c)** \( \frac{1}{8} \) - **(d)** \( \frac{9}{4} \)

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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) \( \frac{9}{4} \)

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