To solve the expression [tex]$\left(3^r\right)^2$[/tex], we need to follow these steps:
1. Understand the expression: We are dealing with an exponential expression where a base raised to a power [tex]$r$[/tex] is further squared.
2. Apply the power rule of exponents: When you have an expression of the form [tex]\((a^m)^n\)[/tex], it simplifies to [tex]\(a^{m \cdot n}\)[/tex]. In this case, our base [tex]\(a\)[/tex] is 3, [tex]\(m\)[/tex] is [tex]\(r\)[/tex], and [tex]\(n\)[/tex] is 2. Thus, we can rewrite the expression:
[tex]\[
\left(3^r\right)^2 = 3^{r \cdot 2}
\][/tex]
3. Simplify the exponent: Multiply the exponents:
[tex]\[
3^{r \cdot 2} = 3^{2r}
\][/tex]
So, [tex]\(\left(3^r\right)^2\)[/tex] simplifies to [tex]\(3^{2r}\)[/tex].
4. Express the resulting exponentiation in another form (if necessary): Noticed that any expression of the form [tex]\(a^b\)[/tex] can also be written considering the properties of exponents. Here, we have:
[tex]\[
3^{2r}
\][/tex]
This can also be written as:
[tex]\[
\left(3^2\right)^r = 9^r
\][/tex]
Thus, the expression [tex]\(\left(3^r\right)^2\)[/tex] simplifies to two equivalent forms:
[tex]\[
3^{2r} \quad \text{or} \quad 9^r
\][/tex]
