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

Video Solution

Solution Steps

00:00 Solve the following problem

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

00:08 When we have a number (A) to the power of (B) in a root of order (C)

00:13 The result equals number (A) to the root of order of their product (B times C)

00:17 We'll apply this formula to our exercise, and proceed to calculate the order multiplication

00:30 When we have a root of a multiplication (A times B)

00:33 We can write it as a multiplication of the root of each term

00:42 We'll apply this formula to our exercise, and break down the root

00:51 When we have a number (A) to the power of (B) in a root of order (C)

00:54 The result equals number (A) to the power of their quotient (B divided by C)

00:57 We'll apply this formula to our exercise, and proceed to calculate the quotient of powers

01:05 This is the solution

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 $\sqrt[4]{5} \cdot x$ .