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Chapter 7: Problem 103

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

Short Answer

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

Step by step solution

01

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, \(8^{\frac{2}{3}} = 4\).
02

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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Most popular questions from this chapter

Find the product by inspection: \(\sqrt{3}(\mathrm{x}-\sqrt{5})(\mathrm{x}+\sqrt{5})\).

Express the product \(\left(1 /{ }^{3} \sqrt{\mathrm{x}}^{2}\right) \cdot\left(1 /{ }^{4} \sqrt{\mathrm{x}}\right)\) in simplest radical form, rationalizing the denominator.

Simplify the quotient \(\sqrt{x} /^{4} \sqrt{x}\). Write the result in exponential notation.

Find the product \(4 \sqrt{\left(x^{3} y\right)} \cdot{ }^{4} \sqrt{\left(x y^{2}\right)}\) and simplify.

Reduce \((2+3 \sqrt{-1})^{2} /(2+\sqrt{-1})\) to the form \(\mathrm{A}+\mathrm{B} \sqrt{-1}\).
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