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

Video Solution

Solution Steps

00:09 Let's solve this math problem together.

00:12 A regular root is usually of order two, like a square root.

00:19 When you have a number A, raised to the power of B, inside a root of order C.

00:26 The result is A, raised to the power of B divided by C.

00:32 Let's use this formula! We'll calculate by multiplying the orders.

00:40 For a root of a product like A times B.

00:45 We can split it into separate roots for each number.

00:48 Let's use this idea in our exercise to simplify the root.

00:53 We break down eighty-one into three raised to the power of four.

00:59 For a number like four, to the power of four, in a root of order C.

01:05 You get three, to the power of four divided by four.

01:09 Let's calculate these power quotients for our exercise.

01:13 And that's how we find the solution! Great job!

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 $3x$ .