1. The problem is to calculate the fifth root of 8, written as $\sqrt[5]{8}$. 2. Recall that the fifth root of a number $x$ is the number $y$ such that $y^5 = x$. 3. Express 8 as a power of 2: $8 = 2^3$. 4. Rewrite the fifth root using exponents: $\sqrt[5]{8} = 8^{\frac{1}{5}}$. Substitute $8 = 2^3$: $$8^{\frac{1}{5}} = (2^3)^{\frac{1}{5}}.$$ 5. Use the power of a power rule: $(a^m)^n = a^{m \times n}$. So, $$(2^3)^{\frac{1}{5}} = 2^{3 \times \frac{1}{5}} = 2^{\frac{3}{5}}.$$ 6. The exact simplified form of $\sqrt[5]{8}$ is $2^{\frac{3}{5}}$. If a decimal approximation is needed, $2^{\frac{3}{5}} \approx 1.5157$. Therefore, $\boxed{\sqrt[5]{8} = 2^{\frac{3}{5}} \approx 1.5157}$.