Question
If the expression 5^10 was simplified using the Power Rule of Exponents, which of the following would be an expanded equivalent expression? Show your work. Responses (5⋅5)8 open paren 5 times 5 close paren to the 8th power - no response given (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) open paren 5 times 5 times 5 times 5 times 5 close paren times open paren 5 times 5 times 5 times 5 times 5 close paren times open paren 5 times 5 times 5 times 5 times 5 close paren times open paren 5 times 5 times 5 times 5 times 5 close paren times open paren 5 times 5 times 5 times 5 times 5 close paren - no response given (516)6 open paren 5 to the 16th power close paren to the 6th power - no response given (5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)
Answers
GPT-4o mini
To find an expanded equivalent expression for \(5^{10}\), we will use the properties of exponents, specifically understanding how exponentiation works.
The expression \(5^{10}\) means that the base (5) is multiplied by itself 10 times. Thus, it can be expressed as:
\[
5^{10} = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5
\]
Now, let's analyze the provided options to identify which is an equivalent expanded form.
1. **(5⋅5)8:** This means \( (5 \cdot 5)^8 \). By the Power Rule, \( (5^2)^8 = 5^{16} \). This does not equal \(5^{10}\).
2. **(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):** This is multiplying five groups of five terms, each having five \(5\)s, which results in \(5^{25}\) (since \(5 \times 5 = 5^2\), and \(5^2 \times 5^2 \times 5^2 \times 5^2 \times 5^2 = 5^{10}\)). Thus, this is too high.
3. **(516)6:** This means \((5^{16})^6\). Using the power of a power property, this simplifies to \(5^{96}\), which is also too high.
4. **(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5):** This means multiplying together five groups of \( (5 \cdot 5) \). Each \( (5 \cdot 5) = 5^2\), and therefore:
\[
(5 \cdot 5)^{5} = (5^2)^{5} = 5^{10}
\]
This matches our original term \(5^{10}\).
So, the correct answer is:
\[
\text{(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)}
\]
The expression \(5^{10}\) means that the base (5) is multiplied by itself 10 times. Thus, it can be expressed as:
\[
5^{10} = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5
\]
Now, let's analyze the provided options to identify which is an equivalent expanded form.
1. **(5⋅5)8:** This means \( (5 \cdot 5)^8 \). By the Power Rule, \( (5^2)^8 = 5^{16} \). This does not equal \(5^{10}\).
2. **(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):** This is multiplying five groups of five terms, each having five \(5\)s, which results in \(5^{25}\) (since \(5 \times 5 = 5^2\), and \(5^2 \times 5^2 \times 5^2 \times 5^2 \times 5^2 = 5^{10}\)). Thus, this is too high.
3. **(516)6:** This means \((5^{16})^6\). Using the power of a power property, this simplifies to \(5^{96}\), which is also too high.
4. **(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5):** This means multiplying together five groups of \( (5 \cdot 5) \). Each \( (5 \cdot 5) = 5^2\), and therefore:
\[
(5 \cdot 5)^{5} = (5^2)^{5} = 5^{10}
\]
This matches our original term \(5^{10}\).
So, the correct answer is:
\[
\text{(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)}
\]