Certainly! Let's simplify the expression [tex]\(\frac{3^4}{9}\)[/tex] by using the Quotient of Powers Property step-by-step:
1. Identify the bases and their exponents:
- The base in the numerator is 3 with an exponent of 4, so we have [tex]\(3^4\)[/tex].
- The denominator is 9. Recognize that 9 can be expressed as [tex]\(3^2\)[/tex].
So, the expression becomes:
[tex]\[
\frac{3^4}{9} = \frac{3^4}{3^2}
\][/tex]
2. Apply the Quotient of Powers Property:
The Quotient of Powers Property states that [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], where [tex]\(a\)[/tex] is the base and [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are the exponents.
Using this property, we substitute [tex]\(a = 3\)[/tex], [tex]\(m = 4\)[/tex], and [tex]\(n = 2\)[/tex] into the expression:
[tex]\[
\frac{3^4}{3^2} = 3^{4-2}
\][/tex]
3. Subtract the exponents:
Perform the subtraction of the exponents:
[tex]\[
4 - 2 = 2
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
3^{4-2} = 3^2
\][/tex]
4. Simplify the resulting exponentiation:
Evaluate [tex]\(3^2\)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
So, the expression [tex]\(\frac{3^4}{9}\)[/tex] simplifies to [tex]\(3^2\)[/tex], which equals 9.
To sum up, by expressing the denominator [tex]\(9\)[/tex] as [tex]\(3^2\)[/tex] and then applying the Quotient of Powers Property ([tex]\(\frac{3^4}{3^2}\)[/tex]), we subtracted the exponents, resulting in [tex]\(3^2\)[/tex], which simplifies to [tex]\(9\)[/tex].
