Sure, let's simplify the expression [tex]\( 9^{\frac{3}{2}} \)[/tex] step by step.
1. Understand the Exponent:
The expression [tex]\( 9^{\frac{3}{2}} \)[/tex] can be interpreted using the laws of exponents and radicals. Specifically, [tex]\( a^{m/n} \)[/tex] is equivalent to [tex]\( (a^m)^{1/n} \)[/tex] or [tex]\( \sqrt[n]{a^m} \)[/tex].
2. Break Down the Exponent:
We break [tex]\( 9^{\frac{3}{2}} \)[/tex] into more manageable parts:
[tex]\[
9^{\frac{3}{2}} = (9^3)^{\frac{1}{2}} \quad \text{or} \quad 9^{\frac{1}{2}} \cdot 9^{\frac{3}{2} - \frac{1}{2}}
\][/tex]
Since [tex]\( 9 = 3^2 \)[/tex], we can write:
[tex]\[
9^{\frac{3}{2}} = (3^2)^{\frac{3}{2}}
\][/tex]
3. Apply the Power of a Power Rule:
Using the rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[
(3^2)^{\frac{3}{2}} = 3^{2 \cdot \frac{3}{2}}
\][/tex]
4. Simplify the Exponent Multiplication:
Multiply the exponents:
[tex]\[
2 \cdot \frac{3}{2} = 3
\][/tex]
So the expression simplifies to:
[tex]\[
3^3
\][/tex]
5. Find the Value:
Finally, calculate the value of [tex]\( 3^3 \)[/tex]:
[tex]\[
3^3 = 3 \times 3 \times 3 = 27
\][/tex]
Therefore, the simplified value of [tex]\( 9^{\frac{3}{2}} \)[/tex] is:
[tex]\[
27
\][/tex]
