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Chapter 7: Chapter 7

Problem 103

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Find the numerical value of each of the following. (a) \(8^{2 / 3}\) (b) \(25^{3 / 2}\)

Short Answer

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The numerical values for the given expressions are: a) \(8^{\frac{2}{3}} = 4\) b) \(25^{\frac{3}{2}} = 125\)
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Solve for \(8^{\frac{2}{3}}\)

First, we need to find the value of \(8^{\frac{2}{3}}\). Using the property \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\), we can rewrite the expression as: \(\sqrt[3]{8^2}\). Now let's calculate: 1. Compute the numerator (power): \(8^2 = 64\) 2. Find the cube root (denominator is 3): \(\sqrt[3]{64} = 4\) So, \(8^{\frac{2}{3}} = 4\).

Step 2: Solve for \(25^{\frac{3}{2}}\)

Next, we need to find the value of \(25^{\frac{3}{2}}\). Using the property \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\), we can rewrite the expression as: \(\sqrt[2]{25^3}\). Since the denominator is 2, we know it represents the square root. Now let's calculate: 1. Compute the numerator (power): \(25^3 = 15625\) 2. Find the square root (denominator is 2): \(\sqrt{15625} = 125\) So, \(25^{\frac{3}{2}} = 125\). The numerical values for the given expressions are: a) \(8^{\frac{2}{3}} = 4\) b) \(25^{\frac{3}{2}} = 125\)

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