Question

Use the Product Rule of
Exponents to simplify
(1 point)
5 ⋅
10 5
5
5
15
25
50
5
50
25
15

Answer 1

To simplify the expression \( 5 \cdot 10^5 \cdot 5^{15} \cdot 25^{50} \), we can use the Product Rule of Exponents, which states that \( a^m \cdot a^n = a^{m+n} \).

First, let's rewrite \( 10^5 \) and \( 25^{50} \) in terms of base 5.

We know:
- \( 10 = 2 \cdot 5 \), so \( 10^5 = (2 \cdot 5)^5 = 2^5 \cdot 5^5 \).
- \( 25 = 5^2 \), so \( 25^{50} = (5^2)^{50} = 5^{100} \).

Now we can rewrite the entire expression:

\[
5 \cdot 10^5 \cdot 5^{15} \cdot 25^{50} = 5 \cdot (2^5 \cdot 5^5) \cdot 5^{15} \cdot 5^{100}.
\]

Now, combining the terms with base \(5\):

\[
= 5^1 \cdot 5^5 \cdot 5^{15} \cdot 5^{100} \cdot 2^5
= 2^5 \cdot 5^{1 + 5 + 15 + 100}.
\]

Adding the exponents together:

\[
1 + 5 + 15 + 100 = 121.
\]

So we have:

\[
= 2^5 \cdot 5^{121}.
\]

Finally, the simplified form of the expression is:

\[
2^5 \cdot 5^{121}.
\]