Question

Use the Product Rule of Exponents to simplify 3^5⋅3^7 show you're work

Answer 1

To simplify the expression \( 3^5 \cdot 3^7 \) using the Product Rule of Exponents, we can follow these steps:

The Product Rule of Exponents states that when you multiply two expressions that have the same base, you can add the exponents. Mathematically, it's expressed as:

\[
a^m \cdot a^n = a^{m+n}
\]

In our case, we have:

- Base \( a = 3 \)
- Exponent \( m = 5 \)
- Exponent \( n = 7 \)

Now, applying the Product Rule:

\[
3^5 \cdot 3^7 = 3^{5+7}
\]

Now, we sum the exponents:

\[
5 + 7 = 12
\]

So we have:

\[
3^5 \cdot 3^7 = 3^{12}
\]

Thus, the simplified form of \( 3^5 \cdot 3^7 \) is:

\[
\boxed{3^{12}}
\]