To simplify the given expression [tex]\(\frac{3^4 a^7 b^3}{9 a^2 b^5}\)[/tex], we will follow these steps:
1. Simplify the numerical part:
- The numerator is [tex]\(3^4\)[/tex].
- The denominator is [tex]\(9\)[/tex], which can be rewritten as [tex]\(3^2\)[/tex].
Therefore, the numerical part of the fraction can be rewritten as:
[tex]\[
\frac{3^4}{3^2}
\][/tex]
Using exponent laws [tex]\( \frac{3^m}{3^n} = 3^{m-n} \)[/tex]:
[tex]\[
\frac{3^4}{3^2} = 3^{4-2} = 3^2
\][/tex]
Hence,
[tex]\[
3^2 = 9
\][/tex]
2. Simplify the variable part:
- For [tex]\(a\)[/tex], the power in the numerator is [tex]\(a^7\)[/tex] and in the denominator is [tex]\(a^2\)[/tex]. Using similar exponent laws:
[tex]\[
\frac{a^7}{a^2} = a^{7-2} = a^5
\][/tex]
- For [tex]\(b\)[/tex], the power in the numerator is [tex]\(b^3\)[/tex] and in the denominator is [tex]\(b^5\)[/tex]. Again, using exponent laws:
[tex]\[
\frac{b^3}{b^5} = b^{3-5} = b^{-2}
\][/tex]
3. Combine the simplified components:
- The simplified numerical part is 9.
- The simplified expression for [tex]\(a\)[/tex] is [tex]\(a^5\)[/tex].
- The simplified expression for [tex]\(b\)[/tex] is [tex]\(b^{-2}\)[/tex].
Therefore, the simplified form of the expression is:
[tex]\[
9 a^5 b^{-2}
\][/tex]
