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Chapter 10: Problem 18

Explain how to write \(100^{3 / 2}\) in radical form.

Short Answer

Expert verified
\(100^{\frac{3}{2}}\) can be rewritten as a radical expression: \((\sqrt[2]{100})^3\). Simplifying the square root and calculating the final result, we have \((10)^3 = 1{,}000\).

Step by step solution

01

Recall the rules of exponents and radicals

A general exponential expression is of the form \(a^{\frac{m}{n}}\), where a is the base and m/n is a fractional exponent. To rewrite this in radical form, we can think of the exponent as indicating the the power of m and the root of n. So, \(a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m\) Additionally, we should recall the following radicals: \( \sqrt{a} = a^{\frac{1}{2}}\) for the square root of a \(\sqrt[3]{a} = a^{\frac{1}{3}}\) for the cube root of a
02

Rewrite the given expression as a radical expression

To rewrite the given expression, \(100^{\frac{3}{2}}\), as a radical expression, we will apply the rules we discussed earlier: \( 100^{\frac{3}{2}} = \sqrt[2]{100^3} = (\sqrt[2]{100})^3 \)
03

Simplify the expression

Now that the expression is in radical form, let's simplify the square root: \( \sqrt[2]{100} = \sqrt{100} = 10 \) So our expression in simplified form is: \( (\sqrt[2]{100})^3 = (10)^3 \)
04

Calculate the final result

Finally, calculate \(10^3\): \(10^3 = 10 \times 10 \times 10 = 1{,}000\) So, \(100^{\frac{3}{2}}\) in radical form is \(1{,}000\).

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