The problem asks us to simplify $49^{\frac{3}{2}}$.

1. Problem Description

The problem asks us to simplify $49^{\frac{3}{2}}$.

2. Solution Steps

We can rewrite $49^{\frac{3}{2}}$ as $(49^{\frac{1}{2}})^3$ or $(49^3)^{\frac{1}{2}}$. It's easier to first take the square root.

$49^{\frac{3}{2}} = (49^{\frac{1}{2}})^3$

We know that $49^{\frac{1}{2}}$ is the square root of

4

9.

$49^{\frac{1}{2}} = \sqrt{49} = 7$

So,

$49^{\frac{3}{2}} = (7)^3$

Now we calculate $7^3$:

$7^3 = 7 \cdot 7 \cdot 7 = 49 \cdot 7 = 343$

Therefore, $49^{\frac{3}{2}} = 343$.

3. Final Answer

343