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

Video Solution

Solution Steps

00:00 Solve the following problem

00:03 A "regular" root is of the order 2

00:11 When we have a number (A) to the power of (B) in a root of order (C)

00:14 The result equals the number (A) to the power of their quotient (B times C)

00:18 We will apply this formula to our exercise

00:24 Calculate the order multiplication

00:31 When we have a root of a product (A times B)

00:36 We can write it as the product of the root of each term

00:40 We will apply this formula to our exercise, and proceed to break down the root

00:49 When we have a number (A) to the power of (B) in a root of order (C)

00:52 The result equals the number (A) to the power of their quotient (B divided by C)

00:56 We will apply this formula to our exercise

01:04 Calculate the power quotient

01:12 We'll apply this formula again in the opposite direction, converting from a power to a root

01:22 This is the solution

Step-by-Step Solution

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 $\sqrt[4]{3} \cdot \sqrt{x}$ .