Question

If the expression 5^10 was simplified using the power rule of exponents, which if the following would be an expanded equivalent expression?
(5*5) * (5*5) * (5*5) * (5*5) * (5*5)
(5*5)^8
(5^16)^6
(5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5)

Answer 1

To determine which of the given options is an expanded equivalent expression for \(5^{10}\), we'll first analyze what \(5^{10}\) means and how the power rule of exponents applies.

The expression \(5^{10}\) represents multiplying \(5\) by itself a total of 10 times:

\[
5^{10} = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5
\]

Now let's examine each option:

1. **\((5*5) * (5*5) * (5*5) * (5*5) * (5*5)\)**
- This breaks down into:
\[
(5^2) * (5^2) * (5^2) * (5^2) * (5^2)
\]
- Since \(5^2 = 25\), we have \(5 \times 5 = 5^2\). Therefore, when we multiply five \(5^2\) terms:
\[
(5^2)^5 = 5^{2 \cdot 5} = 5^{10}
\]
- This option is equivalent to \(5^{10}\).

2. **\((5*5)^8\)**
- Simplifying gives us:
\[
(5^2)^8 = 5^{2 \cdot 8} = 5^{16}
\]
- This is not equivalent to \(5^{10}\).

3. **\((5^{16})^6\)**
- Simplifying gives us:
\[
5^{16 \cdot 6} = 5^{96}
\]
- This is not equivalent to \(5^{10}\).

4. **\((5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5)\)**
- There are five groups of \(5\) multiplied together (each group being \(5^5\)):
\[
(5^5) * (5^5) * (5^5) * (5^5) * (5^5)
\]
- This results in:
\[
(5^5)^5 = 5^{5 \cdot 5} = 5^{25}
\]
- This is not equivalent to \(5^{10}\).

The only expression that accurately simplifies to \(5^{10}\) is the first option:

\[
(5*5) * (5*5) * (5*5) * (5*5) * (5*5)
\]

Thus, the expanded equivalent expression for \(5^{10}\) is **Option 1**.