1. The problem states that the cube root of $a$ equals the square root of 5, i.e., $\sqrt[3]{a} = \sqrt{5}$. 2. We want to find the value of $a$. To do this, we can cube both sides of the equation to eliminate the cube root. 3. Cubing both sides gives: $$\left(\sqrt[3]{a}\right)^3 = \left(\sqrt{5}\right)^3$$ 4. Simplifying the left side, the cube and cube root cancel out, leaving: $$a = \left(\sqrt{5}\right)^3$$ 5. Now, write the right side with exponents: $$a = (5^{1/2})^3 = 5^{3/2}$$ 6. This can be further simplified as: $$a = 5^{1 + \frac{1}{2}} = 5^{1.5} = 5 \times 5^{0.5} = 5 \times \sqrt{5}$$ 7. Therefore, the value of $a$ is: $$a = 5 \sqrt{5}$$