Simplify the Nested Square Root: √(√(3x²))

Complete the following exercise:

$\sqrt{\sqrt{3x^2}}=$

To solve $\sqrt{\sqrt{3x^2}}$, follow these steps:

• Step 1: Express the problem using exponentiation. The expression $\sqrt{3x^2}$ can be written as $(3x^2)^{\frac{1}{2}}$.
• Step 2: Take the square root of the first expression. This can be expressed as $((3x^2)^{\frac{1}{2}})^{\frac{1}{2}}$.
• Step 3: Use the property $(a^m)^n = a^{m \cdot n}$. Thus, $((3x^2)^{\frac{1}{2}})^{\frac{1}{2}} = (3x^2)^{\frac{1}{4}}$.
• Step 4: Simplify further using exponent rules: $(3x^2)^{\frac{1}{4}}$ becomes $(3^{\frac{1}{4}} \cdot (x^2)^{\frac{1}{4}})$, which simplifies to $\sqrt[4]{3} \cdot x^{\frac{1}{2}}$.
• Step 5: Recognize this as $\sqrt[4]{3} \cdot \sqrt{x}$, since $x^{\frac{1}{2}}$ is $\sqrt{x}$.

Therefore, the simplified form of the given expression is $\sqrt[4]{3} \cdot \sqrt{x}$.