Answer :
To determine which choice is equivalent to the product [tex]\(\sqrt{5 x^2} \cdot \sqrt{15 x^2}\)[/tex] for [tex]\(x \geq 0\)[/tex], let's break down the expression step by step.
1. First, consider the properties of radicals. The product of two square roots can be combined into a single square root:
[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]
Using this property, we can combine our original expression:
[tex]\[ \sqrt{5 x^2} \cdot \sqrt{15 x^2} = \sqrt{(5 x^2) \cdot (15 x^2)} \][/tex]
2. Next, multiply the expressions inside the square root:
[tex]\[ (5 x^2) \cdot (15 x^2) = 5 \cdot 15 \cdot x^2 \cdot x^2 = 75 x^4 \][/tex]
So, we have:
[tex]\[ \sqrt{75 x^4} \][/tex]
3. Now simplify [tex]\(\sqrt{75 x^4}\)[/tex]. Notice that [tex]\(75 = 3 \cdot 25\)[/tex], which can be written as [tex]\(5^2 \cdot 3\)[/tex]:
[tex]\[ \sqrt{75 x^4} = \sqrt{3 \cdot 5^2 \cdot x^4} \][/tex]
4. Since [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex] applies here, we can split the square root:
[tex]\[ \sqrt{75 x^4} = \sqrt{3} \cdot \sqrt{5^2} \cdot \sqrt{x^4} \][/tex]
5. Simplify each square root:
[tex]\[ \sqrt{5^2} = 5 \quad \text{and} \quad \sqrt{x^4} = x^2 \][/tex]
Therefore:
[tex]\[ \sqrt{75 x^4} = 5 \sqrt{3} x^2 \][/tex]
Finally, the expression [tex]\(\sqrt{5 x^2} \cdot \sqrt{15 x^2}\)[/tex] simplifies to [tex]\(5 \sqrt{3} x^2\)[/tex]. Thus, the equivalent choice is:
[tex]\[ \boxed{5 x^2 \sqrt{3}} \][/tex]
1. First, consider the properties of radicals. The product of two square roots can be combined into a single square root:
[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]
Using this property, we can combine our original expression:
[tex]\[ \sqrt{5 x^2} \cdot \sqrt{15 x^2} = \sqrt{(5 x^2) \cdot (15 x^2)} \][/tex]
2. Next, multiply the expressions inside the square root:
[tex]\[ (5 x^2) \cdot (15 x^2) = 5 \cdot 15 \cdot x^2 \cdot x^2 = 75 x^4 \][/tex]
So, we have:
[tex]\[ \sqrt{75 x^4} \][/tex]
3. Now simplify [tex]\(\sqrt{75 x^4}\)[/tex]. Notice that [tex]\(75 = 3 \cdot 25\)[/tex], which can be written as [tex]\(5^2 \cdot 3\)[/tex]:
[tex]\[ \sqrt{75 x^4} = \sqrt{3 \cdot 5^2 \cdot x^4} \][/tex]
4. Since [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex] applies here, we can split the square root:
[tex]\[ \sqrt{75 x^4} = \sqrt{3} \cdot \sqrt{5^2} \cdot \sqrt{x^4} \][/tex]
5. Simplify each square root:
[tex]\[ \sqrt{5^2} = 5 \quad \text{and} \quad \sqrt{x^4} = x^2 \][/tex]
Therefore:
[tex]\[ \sqrt{75 x^4} = 5 \sqrt{3} x^2 \][/tex]
Finally, the expression [tex]\(\sqrt{5 x^2} \cdot \sqrt{15 x^2}\)[/tex] simplifies to [tex]\(5 \sqrt{3} x^2\)[/tex]. Thus, the equivalent choice is:
[tex]\[ \boxed{5 x^2 \sqrt{3}} \][/tex]
