To simplify the expression [tex]\(\left(p^2 q^3\right)^2\)[/tex], we can use the laws of exponents.
Step-by-Step Solution:
1. Identify the expression inside the parentheses:
[tex]\[
p^2 q^3
\][/tex]
Here, [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are variables, and both are raised to specific powers (2 for [tex]\(p\)[/tex] and 3 for [tex]\(q\)[/tex]).
2. Apply the exponentiation rule [tex]\((a^m b^n)^k = a^{mk} b^{nk}\)[/tex]:
[tex]\[
\left(p^2 q^3\right)^2
\][/tex]
According to the rule, each term inside the parentheses must be raised to the power of 2.
3. Distribute the outer exponent to both [tex]\(p\)[/tex] and [tex]\(q\)[/tex]:
[tex]\[
\left(p^2\right)^2 \left(q^3\right)^2
\][/tex]
4. Simplify each term separately using the power of a power rule [tex]\((a^m)^k = a^{m \cdot k}\)[/tex]:
[tex]\[
(p^2)^2 = p^{2 \cdot 2} = p^4
\][/tex]
[tex]\[
(q^3)^2 = q^{3 \cdot 2} = q^6
\][/tex]
5. Combine the simplified terms:
[tex]\[
p^4 q^6
\][/tex]
So, the simplified form of the expression [tex]\(\left(p^2 q^3\right)^2\)[/tex] is:
[tex]\[
p^4 q^6
\][/tex]
