To simplify the expression [tex]\( 9^{\frac{3}{2}} \)[/tex], let's follow these steps:
1. Represent the Base as a Power:
The base 9 can be written as a power of 3 because [tex]\( 9 = 3^2 \)[/tex]. Therefore:
[tex]\[
9^{\frac{3}{2}} = (3^2)^{\frac{3}{2}}
\][/tex]
2. Apply the Exponentiation Rule:
When raising a power to another power, you multiply the exponents. Thus:
[tex]\[
(3^2)^{\frac{3}{2}} = 3^{2 \cdot \frac{3}{2}}
\][/tex]
3. Simplify the Exponents:
Now we multiply the exponents:
[tex]\[
2 \cdot \frac{3}{2} = 3
\][/tex]
4. Evaluate the Simplified Expression:
So now our expression becomes:
[tex]\[
3^3
\][/tex]
5. Calculate the Final Power:
Finally, we evaluate [tex]\( 3^3 \)[/tex]:
[tex]\[
3^3 = 3 \times 3 \times 3 = 27
\][/tex]
Therefore, the simplified form of [tex]\( 9^{\frac{3}{2}} \)[/tex] is:
[tex]\[
27
\][/tex]
