To simplify the expression [tex]\(\frac{\left(a^3 a^6\right)^{\frac{1}{4}}}{a^2}\)[/tex], we can follow these steps:
1. Simplify the expression inside the parentheses:
[tex]\[
a^3 \cdot a^6
\][/tex]
Here, we use the property of exponents that states when multiplying like bases, we add the exponents. So,
[tex]\[
a^3 \cdot a^6 = a^{3+6} = a^9
\][/tex]
2. Rewrite the expression with the simplified base:
[tex]\[
\frac{\left(a^9\right)^{\frac{1}{4}}}{a^2}
\][/tex]
3. Apply the power rule to the numerator:
The power rule for exponents states that [tex]\((a^m)^n = a^{mn}\)[/tex]. Therefore,
[tex]\[
\left(a^9\right)^{\frac{1}{4}} = a^{9 \cdot \frac{1}{4}} = a^{\frac{9}{4}}
\][/tex]
4. Rewrite the expression with the simplified numerator:
[tex]\[
\frac{a^{\frac{9}{4}}}{a^2}
\][/tex]
5. Simplify the division of exponents:
When dividing like bases, we subtract the exponents:
[tex]\[
a^{\frac{9}{4}} \div a^2 = a^{\frac{9}{4} - 2}
\][/tex]
Convert 2 to a fraction with the same denominator as [tex]\(\frac{9}{4}\)[/tex]:
[tex]\[
2 = \frac{8}{4}
\][/tex]
6. Subtract the exponents:
[tex]\[
\frac{9}{4} - \frac{8}{4} = \frac{1}{4}
\][/tex]
7. Write the final simplified expression:
[tex]\[
a^{\frac{1}{4}}
\][/tex]
Thus, the simplified expression is:
[tex]\[
a^{\frac{1}{4}}
\][/tex]
