To determine which of the given expressions equal [tex]\(3^8\)[/tex], let's analyze each one step by step.
### Expression A: [tex]\((3^2)^4\)[/tex]
Using the properties of exponents, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(3^2)^4 = 3^{2 \cdot 4} = 3^8
\][/tex]
### Expression: [tex]\(3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3\)[/tex]
The expression is a product of eight 3’s:
[tex]\[
3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 3^8
\][/tex]
### Expression: [tex]\(\left(3^4\right)^2\)[/tex]
Using the properties of exponents again, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(3^4)^2 = 3^{4 \cdot 2} = 3^8
\][/tex]
### Expression E: [tex]\(\frac{3^6}{3^{-2}}\)[/tex]
Using the properties of exponents, [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{3^6}{3^{-2}} = 3^{6 - (-2)} = 3^{6 + 2} = 3^8
\][/tex]
### Expression: [tex]\(3^6 \cdot 10^2\)[/tex]
This expression cannot be simplified to [tex]\(3^8\)[/tex] because the presence of [tex]\(10^2\)[/tex] indicates multiplication by a different base:
[tex]\[
3^6 \cdot 10^2 \text{ does not equal } 3^8
\][/tex]
By analyzing these expressions, we see that the expressions equal to [tex]\(3^8\)[/tex] are:
1. [tex]\((3^2)^4\)[/tex]
2. [tex]\(3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3\)[/tex]
3. [tex]\((3^4)^2\)[/tex]
4. [tex]\(\frac{3^6}{3^{-2}}\)[/tex]
Hence, the four expressions which equal [tex]\(3^8\)[/tex] are:
- [tex]\((3^2)^4\)[/tex]
- [tex]\(3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3\)[/tex]
- [tex]\((3^4)^2\)[/tex]
- [tex]\(\frac{3^6}{3^{-2}}\)[/tex]
These correspond to the expressions A, the product of 3's, the exponents of 3's in the numerator and denominator, and [tex]\((3^4)^2\)[/tex] found from the given options.
