1. The problem asks us to find the value of $49^{\frac{3}{2}}$. 2. The exponent $\frac{3}{2}$ means we take the square root (denominator 2) and then cube the result (numerator 3), or vice versa. 3. Using the rule $a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$, we can write: $$49^{\frac{3}{2}} = \left(\sqrt{49}\right)^3$$ 4. Calculate the square root of 49: $$\sqrt{49} = 7$$ 5. Now cube 7: $$7^3 = 7 \times 7 \times 7 = 343$$ 6. Therefore, the value of $49^{\frac{3}{2}}$ is $343$. This method uses the properties of fractional exponents to simplify the expression step-by-step.