Simplify the Nested Square Root Expression: √(√(5x⁴))

Video Solution

Solution Steps

00:07 Let's solve this problem together.

00:10 A regular square root is of order two.

00:15 If we have A to the power of B inside a root of order C,

00:20 It equals A to the power of B times C.

00:25 Let's use this formula and multiply the powers in our exercise.

00:37 With a root of A times B,

00:40 We can split it into the root of A times the root of B.

00:49 Now we'll apply this rule, and separate the roots in our problem.

00:58 If A to the power of B is in a root of order C,

01:02 It becomes A to the power of B divided by C.

01:06 We'll use this to divide the exponents in our task.

01:12 And that's how we solve this problem.

Step-by-Step Solution

To solve the expression $\sqrt[]{\sqrt{5x^4}}$ , let's go step-by-step:

Step 1: Simplify the inner expression $\sqrt{5x^4}$ . Using the rule for square roots, we can rewrite $\sqrt{5x^4}$ as $(5x^4)^{1/2}$ . This expression can be further simplified to $5^{1/2} \cdot (x^4)^{1/2} = \sqrt{5} \cdot x^2$ .
Step 2: Take the square root of the simplified expression. This means we apply another square root to $\sqrt{5} \cdot x^2$ , resulting in $(\sqrt{5} \cdot x^2)^{1/2} = (\sqrt{5})^{1/2} \cdot (x^2)^{1/2}$ .
Step 3: Simplify each component: $\sqrt[4]{5} \cdot x$ . We find that $(\sqrt{5})^{1/2}$ simplifies to $\sqrt[4]{5}$ and $(x^2)^{1/2}$ to $x$ .

Therefore, the simplified expression is $\sqrt[4]{5} \cdot x$ .