To simplify the expression \(5^{10}\) using the Power Rule of Exponents, we know that the Power Rule states that \(a^{m+n} = a^m \cdot a^n\).
However, in this case, it is more straightforward to recognize that \(5^{10}\) can be expressed by repeated multiplication.
Expansion of \(5^{10}\)
The expression \(5^{10}\) can be expanded as follows:
\[ 5^{10} = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \]
This means you can group the 5s into products:
An equivalent expanded form could also group them as pairs or any arrangement as long as you have a total of ten 5s multiplied together:
- Group of Fives:
- \( (5 \cdot 5) \cdot (5 \cdot 5) \cdot (5 \cdot 5) \cdot (5 \cdot 5) \cdot (5 \cdot 5) \)
This says you have five pairs of \(5 \cdot 5\), which would actually be \(5^2\) each.
Let's Compare with the Provided Responses:
-
Response 1: \((5 \cdot 5)^{8}\)
- This is \(5^{16}\), not \(5^{10}\).
-
Response 2: \((5 \cdot 5 \cdot 5 \cdot 5 \cdot 5) \cdot (5 \cdot 5 \cdot 5 \cdot 5 \cdot 5) \cdot (5 \cdot 5 \cdot 5 \cdot 5 \cdot 5) \cdot (5 \cdot 5 \cdot 5 \cdot 5 \cdot 5) \cdot (5 \cdot 5 \cdot 5 \cdot 5 \cdot 5)\)
- This forms \( (5^5)^{5}\) or \(5^{25}\), not \(5^{10}\).
-
Response 3: \((5^{16})^{6}\)
- This is \(5^{96}\), not \(5^{10}\).
-
Response 4: \((5 \cdot 5) \cdot (5 \cdot 5) \cdot (5 \cdot 5) \cdot (5 \cdot 5) \cdot (5 \cdot 5)\)
- This says \( (5^2)^{5} = 5^{10}\)
- This matches the originally given expression \(5^{10}\).
Given the comparisons, the correct expanded equivalent expression for \(5^{10}\) is:
\[ (5 \cdot 5) \cdot (5 \cdot 5) \cdot (5 \cdot 5) \cdot (5 \cdot 5) \cdot (5 \cdot 5) \]
This would be the Response 4.