Sure, let's break down the problem step-by-step to check which expressions are equivalent to [tex]\(3^2 \cdot 3^4\)[/tex].
Step 1: Evaluate the given expression
[tex]\[3^2 \cdot 3^4\][/tex]
According to the properties of exponents, specifically the Product of Powers Property:
[tex]\[a^m \cdot a^n = a^{m+n}\][/tex]
Thus,
[tex]\[3^2 \cdot 3^4 = 3^{2+4} = 3^6\][/tex]
So, [tex]\(3^2 \cdot 3^4\)[/tex] simplifies to [tex]\(3^6\)[/tex].
Step 2: Compare with each given expression
1. [tex]\(3^6\)[/tex]
[tex]\[3^6\][/tex] is equivalent to [tex]\(3^6\)[/tex].
2. [tex]\(3^8\)[/tex]
[tex]\[3^8\][/tex] is not equivalent to [tex]\(3^6\)[/tex].
3. [tex]\(9^5\)[/tex]
[tex]\[9 = 3^2\][/tex]
Thus,
[tex]\[9^5 = (3^2)^5 = 3^{2 \cdot 5} = 3^{10}\][/tex]
[tex]\[3^{10}\][/tex] is not equivalent to [tex]\(3^6\)[/tex].
4. [tex]\(3^{-4} \cdot 3^{10}\)[/tex]
According to the properties of exponents:
[tex]\[3^{-4} \cdot 3^{10} = 3^{-4+10} = 3^6\][/tex]
[tex]\[3^6\][/tex] is equivalent to [tex]\(3^6\)[/tex].
5. [tex]\(3^0\)[/tex], [tex]\(3^8\)[/tex]
[tex]\[3^0 = 1\][/tex]
[tex]\[1 \cdot 3^8 = 3^8\][/tex]
[tex]\[3^8\][/tex] is not equivalent to [tex]\(3^6\)[/tex].
6. [tex]\(3^3 \cdot 3^3\)[/tex]
According to the properties of exponents:
[tex]\[3^3 \cdot 3^3 = 3^{3+3} = 3^6\][/tex]
[tex]\[3^6\][/tex] is equivalent to [tex]\(3^6\)[/tex].
7. (3^2) [tex]\(\cdot(3^4)\)[/tex]
We interpreted this earlier:
[tex]\[(3^2) \cdot (3^4) = 3^{2+4} = 3^6\][/tex]
8. (3^3) [tex]\(\cdot (3^3)\)[/tex]
Similar to an earlier interpretation:
[tex]\[(3^3) \cdot (3^3) = 3^{3+3} = 3^6\][/tex]
Summary:
The expressions equivalent to [tex]\(3^2 \cdot 3^4\)[/tex] are:
[tex]\[3^6, 3^{-4} \cdot 3^{10}, 3^3 \cdot 3^3, (3^2) \cdot (3^4)\][/tex]
