Answer :
Sure, let's solve the given equation step-by-step to determine the value of [tex]\(5^x\)[/tex].
We start with the given equation:
[tex]\[ (3^3)^2 = 9^x \][/tex]
First, simplify the left-hand side of the equation:
[tex]\[ (3^3)^2 \][/tex]
By using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can rewrite this as:
[tex]\[ 3^{3 \cdot 2} = 3^6 \][/tex]
Next, simplify the right-hand side of the equation. Recall that [tex]\(9\)[/tex] can be written as [tex]\(3^2\)[/tex]:
[tex]\[ 9^x = (3^2)^x \][/tex]
Using the same power rule, we rewrite this as:
[tex]\[ (3^2)^x = 3^{2x} \][/tex]
Now we have the equation:
[tex]\[ 3^6 = 3^{2x} \][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[ 6 = 2x \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{6}{2} = 3 \][/tex]
Now that we have found [tex]\(x = 3\)[/tex], we need to determine [tex]\(5^x\)[/tex]:
[tex]\[ 5^x = 5^3 \][/tex]
Calculating [tex]\(5^3\)[/tex]:
[tex]\[ 5^3 = 125 \][/tex]
Therefore, the value of [tex]\(5^x\)[/tex] is:
[tex]\[ \boxed{125} \][/tex]
We start with the given equation:
[tex]\[ (3^3)^2 = 9^x \][/tex]
First, simplify the left-hand side of the equation:
[tex]\[ (3^3)^2 \][/tex]
By using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can rewrite this as:
[tex]\[ 3^{3 \cdot 2} = 3^6 \][/tex]
Next, simplify the right-hand side of the equation. Recall that [tex]\(9\)[/tex] can be written as [tex]\(3^2\)[/tex]:
[tex]\[ 9^x = (3^2)^x \][/tex]
Using the same power rule, we rewrite this as:
[tex]\[ (3^2)^x = 3^{2x} \][/tex]
Now we have the equation:
[tex]\[ 3^6 = 3^{2x} \][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[ 6 = 2x \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{6}{2} = 3 \][/tex]
Now that we have found [tex]\(x = 3\)[/tex], we need to determine [tex]\(5^x\)[/tex]:
[tex]\[ 5^x = 5^3 \][/tex]
Calculating [tex]\(5^3\)[/tex]:
[tex]\[ 5^3 = 125 \][/tex]
Therefore, the value of [tex]\(5^x\)[/tex] is:
[tex]\[ \boxed{125} \][/tex]
