To solve the expression [tex]\(\frac{3^6}{3^4}\)[/tex], follow these detailed steps:
1. Understand the properties of exponents:
We have the same base, which is [tex]\(3\)[/tex], in both the numerator and the denominator. The law of exponents states that when you divide like bases, you subtract the exponents:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
2. Apply the property to the given expression:
Here, [tex]\(a = 3\)[/tex], [tex]\(m = 6\)[/tex], and [tex]\(n = 4\)[/tex]. Substitute these values into the exponent rule:
[tex]\[
\frac{3^6}{3^4} = 3^{6-4}
\][/tex]
3. Perform the subtraction in the exponent:
Subtract [tex]\(4\)[/tex] from [tex]\(6\)[/tex]:
[tex]\[
6 - 4 = 2
\][/tex]
4. Simplify the expression:
Now we have:
[tex]\[
3^{6-4} = 3^2
\][/tex]
5. Calculate the final result:
Evaluate [tex]\(3^2\)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
Therefore, the result of [tex]\(\frac{3^6}{3^4}\)[/tex] is [tex]\(\boxed{9}\)[/tex].
