Which are equivalent to [tex][tex]$3^2 \cdot 3^4$[/tex][/tex]? Check all that apply.

A. [tex][tex]$3^6$[/tex][/tex]
B. [tex][tex]$3^8$[/tex][/tex]
C. [tex][tex]$9^5$[/tex][/tex]
D. [tex][tex]$3^{-4} \cdot 3^{10}$[/tex][/tex]
E. [tex][tex]$3^0 \cdot 3^8$[/tex][/tex]
F. [tex][tex]$3^3 \cdot 3^3$[/tex][/tex]
G. [tex](3^2) \cdot (3^4)[tex]$[/tex]
H. [tex](3^3) \cdot (3 \cdot 3 \cdot 3 \cdot 3)$[/tex][/tex]



Answer :

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]