Sure, let's go through the steps to simplify the given expression [tex]\(\left(p^2 q^2\right)^2\)[/tex] by applying the laws of indices.
### Step-by-Step Solution
1. Given Expression:
[tex]\[
\left(p^2 q^2\right)^2
\][/tex]
2. Application of the Power Rule:
According to the power rule in exponents:
[tex]\[
(a^m b^n)^k = a^{m \cdot k} \cdot b^{n \cdot k}
\][/tex]
Here, [tex]\(a\)[/tex], [tex]\(b\)[/tex] are the bases, [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are the exponents, and [tex]\(k\)[/tex] is the power to which the entire expression is raised.
3. Applying the Power Rule to Each Factor:
- Identify the bases and their exponents:
[tex]\[
p^2 \quad \text{and} \quad q^2
\][/tex]
- Raise each base to the power outside the parentheses:
[tex]\[
(p^2)^2 \quad \text{and} \quad (q^2)^2
\][/tex]
4. Calculating Individual Exponents:
- For [tex]\(p\)[/tex]:
[tex]\[
(p^2)^2 = p^{2 \cdot 2} = p^4
\][/tex]
- For [tex]\(q\)[/tex]:
[tex]\[
(q^2)^2 = q^{2 \cdot 2} = q^4
\][/tex]
5. Combine the Results:
Put together the simplified forms of each factor:
[tex]\[
(p^2 q^2)^2 = p^4 \cdot q^4
\][/tex]
6. Final Simplified Expression:
[tex]\[
\left(p^2 q^2\right)^2 = p^4 q^4
\][/tex]
So, the simplified form of the expression [tex]\(\left(p^2 q^2\right)^2\)[/tex] is [tex]\(\mathbf{p^4 q^4}\)[/tex].
