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Problem 103

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

Expert verified
The numerical values for the given expressions are: a) $$8^{\frac{2}{3}} = 4$$ b) $$25^{\frac{3}{2}} = 125$$
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## 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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