Solve Nested Square Roots: Simplifying √(√(81x⁴))

Video Solution

Solution Steps

00:00 Solve the following problem

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

00:10 When we have a number (A) to the power of (B) in a root of order (C)

00:16 The result equals number (A) to the power of the quotient (B times C)

00:23 Let's apply this formula to our exercise, and calculate the order multiplication

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

00:34 We can write it as a multiplication of roots of each term

00:37 Let's apply this formula to our exercise, and proceed to break down the root

00:41 Let's break down 81 to 3 to the power of 4

00:50 When we have a number (4) to the power of (4) in a root of order (C)

00:53 The result equals number (3) to the power of their quotient (4 divided by 4)

00:56 Let's apply this formula to our exercise, and calculate the power quotients

00:59 This is the solution

Step-by-Step Solution

To solve the problem $\sqrt{\sqrt{81 \cdot x^4}}$ , we need to simplify this expression using properties of exponents and square roots.

Step 1: Simplify the inner square root
The expression inside the first square root is $81 \cdot x^4$ . We can rewrite this using exponents:
$81 = 9^2$ and $x^4 = (x^2)^2$ . Thus, $81 \cdot x^4 = (9x^2)^2$ .
Step 2: Apply the inner square root
Taking the square root of $(9x^2)^2$ gives us:
$\sqrt{(9x^2)^2} = 9x^2$ , because $\sqrt{a^2} = a$ where $a$ is a non-negative real number.
Step 3: Simplify the outer square root
Now, we take the square root of the result from the inner root:
$\sqrt{9x^2} = \sqrt{9} \cdot \sqrt{x^2} = 3 \cdot x = 3x$ , since $\sqrt{x^2} = x$ given $x$ is non-negative.

Therefore, the solution to the problem is $3x$ .