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Chapter 8: Problem 74

Simplify each expression. Assume that all variables represent positive values. Assume no division by 0. SEE EXAMPLE 6. (OBJECTIVE 3) $$\left(\frac{x^{2 / 9}}{x^{1 / 9}}\right)^{9}$$

Short Answer

Expert verified
The simplified form of the expression \( \left(\frac{x^{2 / 9}}{x^{1 / 9}}\right)^{9} \) is \(x\).

Step by step solution

01

Apply the Quotient Rule of Exponents

The quotient rule of exponents states that \(a^m / a^n = a^{m-n}\). Applying this rule, we get: \(\left(\frac{x^{2 / 9}}{x^{1 / 9}}\right)^{9} = \left(x^{(2/9 - 1/9)}\right)^9 = \left(x^{1/9}\right)^9\)
02

Simplify the Power to a Power

The power of a power property of exponents states that \((a^m)^n = a^{mn}\). Applying this rule, we get: \(\left(x^{1 / 9}\right)^9 = x^{(1/9) * 9} = x^1\)
03

Simplify Final Answer

\(x^1\) simplifies to \(x\). We do not usually write the exponent when it is 1. So the final answer is \(x\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quotient Rule of Exponents
When dealing with exponential expressions such as \( \frac{a^m}{a^n} \) where both the numerator and the denominator have the same base (\

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