Question
Question
If the expression 510
was simplified using the Power Rule of Exponents, which of the following would be an expanded equivalent expression?
Show your work.
(1 point)
Responses
(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)
open paren 5 times 5 close paren times open paren 5 times 5 close paren times open paren 5 times 5 close paren times open paren 5 times 5 close paren times open paren 5 times 5 close paren
(516)6
open paren 5 to the 16th power close paren to the 6th power
(5⋅5)8
open paren 5 times 5 close paren to the 8th power
(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)
If the expression 510
was simplified using the Power Rule of Exponents, which of the following would be an expanded equivalent expression?
Show your work.
(1 point)
Responses
(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)
open paren 5 times 5 close paren times open paren 5 times 5 close paren times open paren 5 times 5 close paren times open paren 5 times 5 close paren times open paren 5 times 5 close paren
(516)6
open paren 5 to the 16th power close paren to the 6th power
(5⋅5)8
open paren 5 times 5 close paren to the 8th power
(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)
Answers
GPT-4o mini
To understand how to simplify the expression \( 5^{10} \) using the Power Rule of Exponents and determine its expanded equivalent expression, we start by recalling the definition of exponentiation.
The expression \( 5^{10} \) means multiplying the base (5) by itself a total of 10 times. This can be represented in its expanded form 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 each option given:
1. **(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)**: This expression simplifies to \( (5^2) \cdot (5^2) \cdot (5^2) \cdot (5^2) \cdot (5^2) \), which equals \( 5^{10} \). This is a correct expanded form.
2. **(516)6**: This expression means \( (5^{16})^6 \), which can be simplified using the Power Rule \( a^{m \cdot n} = a^{m \times n} \). This results in \( 5^{16 \cdot 6} = 5^{96} \). This is not equivalent to \( 5^{10} \).
3. **(5⋅5)8**: This expression is equivalent to \( (5^2)^8 \), which simplifies to \( 5^{2 \cdot 8} = 5^{16} \). This is also 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)**: This expression has 5 groups of \( 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \), or \( 5^5 \), multiplied together. This results in \( (5^5)^5 \), which simplifies to \( 5^{25} \). This is not equivalent to \( 5^{10} \).
Thus, the only correct expanded form for \( 5^{10} \) is the first option:
**(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)**.
The expression \( 5^{10} \) means multiplying the base (5) by itself a total of 10 times. This can be represented in its expanded form 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 each option given:
1. **(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)**: This expression simplifies to \( (5^2) \cdot (5^2) \cdot (5^2) \cdot (5^2) \cdot (5^2) \), which equals \( 5^{10} \). This is a correct expanded form.
2. **(516)6**: This expression means \( (5^{16})^6 \), which can be simplified using the Power Rule \( a^{m \cdot n} = a^{m \times n} \). This results in \( 5^{16 \cdot 6} = 5^{96} \). This is not equivalent to \( 5^{10} \).
3. **(5⋅5)8**: This expression is equivalent to \( (5^2)^8 \), which simplifies to \( 5^{2 \cdot 8} = 5^{16} \). This is also 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)**: This expression has 5 groups of \( 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \), or \( 5^5 \), multiplied together. This results in \( (5^5)^5 \), which simplifies to \( 5^{25} \). This is not equivalent to \( 5^{10} \).
Thus, the only correct expanded form for \( 5^{10} \) is the first option:
**(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)**.