Simplify Square Root 7C070A

1. **State the problem:** Simplify the expression $$\sqrt{75v^{14}}$$. 2. **Recall the formula:** The square root of a product is the product of the square roots: $$\sqrt{ab} = \sqrt{a} \times \sqrt{b}$$. 3. **Apply the formula:** $$\sqrt{75v^{14}} = \sqrt{75} \times \sqrt{v^{14}}$$ 4. **Simplify each part:** - Factor 75 into prime factors: $$75 = 25 \times 3$$. - So, $$\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$$. - For $$\sqrt{v^{14}}$$, since $$\sqrt{v^{14}} = v^{14/2} = v^7$$. 5. **Combine the simplified parts:** $$\sqrt{75v^{14}} = 5\sqrt{3} \times v^7 = 5v^7\sqrt{3}$$. **Final answer:** $$5v^7\sqrt{3}$$