Sure! Let’s simplify the given expression step by step:
Given expression:
[tex]\[ \left[\left(x^{4n - m}\right)\left(\frac{1}{x}\right)\right]^6 \][/tex]
### Step 1: Simplify the inner expression [tex]\(\left(x^{4n - m}\right)\left(\frac{1}{x}\right)\)[/tex]
First, we simplify the multiplication inside the brackets.
The expression [tex]\(\left(x^{4n - m}\right)\left(\frac{1}{x}\right)\)[/tex] can be thought of as:
[tex]\[ x^{4n - m} \cdot x^{-1} \][/tex]
### Step 2: Use the properties of exponents
When multiplying expressions with the same base, we add the exponents:
[tex]\[
x^{4n - m} \cdot x^{-1} = x^{(4n - m) + (-1)} = x^{4n - m - 1}
\][/tex]
### Step 3: Raise the simplified expression to the power of 6
Now we need to raise [tex]\(x^{4n - m - 1}\)[/tex] to the power of 6. This gives us:
[tex]\[
\left(x^{4n - m - 1}\right)^6
\][/tex]
### Step 4: Use the power of a power property
When raising a power to a power, we multiply the exponents:
[tex]\[
\left(x^{4n - m - 1}\right)^6 = x^{(4n - m - 1) \cdot 6}
\][/tex]
### Step 5: Distribute the 6 in the exponent
Distribute the 6 to each term inside the parentheses:
[tex]\[
x^{6 \cdot (4n - m - 1)} = x^{6 \cdot 4n - 6 \cdot m - 6 \cdot 1} = x^{24n - 6m - 6}
\][/tex]
So, the simplified form of the original expression [tex]\(\left[\left(x^{4n - m}\right)\left(\frac{1}{x}\right)\right]^6\)[/tex] is:
[tex]\[
x^{-6m + 24n - 6}
\][/tex]
This is the final answer.
