1. **Stating the problem:** We need to simplify the expression given as $8 \sqrt{584}$. 2. **Formula and rules:** Recall that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ and that we can simplify square roots by factoring out perfect squares. 3. **Factor the number inside the square root:** We want to factor 584 into a product of a perfect square and another number. - Prime factorization of 584: $$584 = 2 \times 292 = 2 \times 2 \times 146 = 2^3 \times 73$$ Since $2^2 = 4$ is a perfect square, we can write: $$584 = 4 \times 146$$ 4. **Simplify the square root:** $$\sqrt{584} = \sqrt{4 \times 146} = \sqrt{4} \times \sqrt{146} = 2 \sqrt{146}$$ 5. **Multiply by 8:** $$8 \sqrt{584} = 8 \times 2 \sqrt{146} = 16 \sqrt{146}$$ 6. **Final answer:** $$8 \sqrt{584} = 16 \sqrt{146}$$ This is the simplified form of the expression.