Questions: Exponents and Exponential Functions Power rules with negative exponents Simplify. [ left(frac2 w^4x^-5right)^-5 ] Write your answer using only positive exponents.

$Exponents and Exponential Functions Power rules with negative exponents Simplify. [ left(frac2 w^4x^-5right)^-5 ] Write your answer using only positive exponents.$

Transcript text: Exponents and Exponential Functions Power rules with negative exponents Simplify. \[ \left(\frac{2 w^{4}}{x^{-5}}\right)^{-5} \] Write your answer using only positive exponents.

Solution

Solution Steps

Step 1: Apply the Negative Exponent Rule

We start with the expression $\left(\frac{2 w^{4}}{x^{-5}}\right)^{-5}$. By applying the negative exponent rule, we can rewrite it as $\left(\frac{x^{-5}}{2 w^{4}}\right)^{5}$.

Step 2: Distribute the Exponent

Next, we distribute the exponent of $5$ to both the numerator and the denominator: \[ \frac{x^{-5 \cdot 5}}{(2 w^{4})^{5}} = \frac{x^{-25}}{2^{5} w^{20}} = \frac{x^{-25}}{32 w^{20}} \]

Step 3: Convert Negative Exponents to Positive

To express the final result with only positive exponents, we convert $x^{-25}$ to $\frac{1}{x^{25}}$: \[ \frac{1}{32 w^{20} x^{25}} \]

Final Answer### Step 1: Apply the Negative Exponent Rule

We start with the expression $\left(\frac{2 w^{4}}{x^{-5}}\right)^{-5}$. By applying the negative exponent rule, we can rewrite it as $\left(\frac{x^{-5}}{2 w^{4}}\right)^{5}$.

Step 2: Distribute the Exponent

Next, we distribute the exponent of $5$ to both the numerator and the denominator: \[ \left(\frac{x^{-5}}{2 w^{4}}\right)^{5} = \frac{(x^{-5})^{5}}{(2 w^{4})^{5}} = \frac{x^{-25}}{32 w^{20}} \]

Step 3: Convert Negative Exponents to Positive

To express the result using only positive exponents, we convert $x^{-25}$ to $\frac{1}{x^{25}}$: \[ \frac{x^{-25}}{32 w^{20}} = \frac{1}{32 w^{20} x^{25}} \]

Final Answer

The simplified expression using only positive exponents is: \[ \boxed{\frac{1}{32 w^{20} x^{25}}} \]

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