To simplify the expression [tex]\(\left(a^4 \cdot b^{-3}\right)^7\)[/tex], let's proceed step-by-step:
### Step 1: Distribute the external exponent to both internal terms
We'll distribute the exponent [tex]\(7\)[/tex] to both [tex]\(a^4\)[/tex] and [tex]\(b^{-3}\)[/tex] inside the parenthesis. This follows the rule [tex]\((x^m \cdot y^n)^p = x^{m \cdot p} \cdot y^{n \cdot p}\)[/tex].
### Step 2: Apply the exponent to each term
For the term [tex]\(a^4\)[/tex]:
[tex]\[ (a^4)^7 = a^{4 \cdot 7} = a^{28} \][/tex]
For the term [tex]\(b^{-3}\)[/tex]:
[tex]\[ (b^{-3})^7 = b^{-3 \cdot 7} = b^{-21} \][/tex]
### Step 3: Write the simplified result
Now that we have the new exponents, we can write the expression as:
[tex]\[ \left(a^4 \cdot b^{-3}\right)^7 = a^{28} \cdot b^{-21} \][/tex]
Thus, the simplified result is:
[tex]\[ a^{28} \cdot b^{-21} \][/tex]
So, the final answer is:
[tex]\[
\left(a^4 \cdot b^{-3}\right)^7 = a^{28} \cdot b^{-21}
\][/tex]
