Certainly! Let's start by breaking the problem down step by step to express the given expression in a simplified rational exponent form.
Given expression:
[tex]\[
\left(\sqrt[3]{\left(x^2-6\right)}\right)^4
\][/tex]
### Step 1: Convert the Cube Root to Exponential Form
The cube root of an expression can be written using a rational exponent. Specifically:
[tex]\[
\sqrt[3]{a} = a^{1/3}
\][/tex]
Applying this to [tex]\(\sqrt[3]{\left(x^2-6\right)}\)[/tex], we get:
[tex]\[
\sqrt[3]{\left(x^2 - 6\right)} = \left(x^2 - 6\right)^{1/3}
\][/tex]
### Step 2: Raise the Result to the Power of 4
Next, we need to raise [tex]\(\left(x^2 - 6\right)^{1/3}\)[/tex] to the power of 4. Using the exponentiation rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can simplify this step.
Thus,
[tex]\[
\left(\left(x^2 - 6\right)^{1/3}\right)^4 = \left(x^2 - 6\right)^{(1/3) \cdot 4}
\][/tex]
### Step 3: Simplify the Exponent
Now, we multiply the exponents:
[tex]\[
\left(x^2 - 6\right)^{(1/3) \cdot 4} = \left(x^2 - 6\right)^{4/3}
\][/tex]
### Result
Hence, the expression [tex]\(\left(\sqrt[3]{\left(x^2-6\right)}\right)^4\)[/tex] in simplified rational exponent form is:
[tex]\[
(x^2 - 6)^{4/3}
\][/tex]
Therefore, the simplified rational exponent form of the original expression is:
[tex]\[
\boxed{(x^2 - 6)^{4/3}}
\][/tex]
There is no need for a file attachment since all the steps are explained here.
