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Chapter 0: Problem 96

Convert the expressions to radical form. $$ \frac{9}{4(1-x)^{-7 / 3}} $$

Short Answer

Expert verified
The given expression in radical form is \( \frac{9}{4}\sqrt[3]{(1-x)^7} \).

Step by step solution

01

Identify the base and exponent

In the given expression, we can identify that the base is \((1 - x)\) and the exponent is a rational number, \(-\frac{7}{3}\).
02

Rewrite the negative exponent as a reciprocal

A negative exponent can be rewritten as a reciprocal with a positive exponent. This means that \((1 - x)^{-7 / 3}\) can be written as: \[ \frac{1}{(1 - x)^{7/3}} \] Now, our goal is to rewrite \((1 - x)^{7 / 3}\) as a radical expression.
03

Rewrite the rational exponent as an equivalent root

To rewrite the rational exponent as an equivalent root, we can write it as the following: \[ (1 - x)^{7 / 3} = \sqrt[3]{(1 - x)^7} \] Here, the exponent 7 comes into the parenthesis, and the root 3 is outside the parenthesis.
04

Substitute the radical expression back in the original expression

Now, we can substitute the radical expression \(\sqrt[3]{(1-x)^7}\) back into the original expression, and get: \[ \frac{9}{4 \cdot \frac{1}{\sqrt[3]{(1-x)^7}}} \]
05

Simplify the expression

Finally, let's simplify the expression by canceling out the reciprocal and multiplying the remaining terms: \[ \frac{9}{4} \cdot \sqrt[3]{(1-x)^7} \] So, the expression in radical form is: \[ \frac{9}{4}\sqrt[3]{(1-x)^7} \]

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