Solve the Nested Radical Equation: √√81 = ∛√x^6

Question

Solve the following exercise:

81=x63 \sqrt{\sqrt{81}}=\sqrt[3]{\sqrt{x^6}}

Video Solution

Solution Steps

00:08 Let's solve this problem together.
00:11 First, break down 81 into 9 to the power of 2.
00:17 Remember, a regular root is like a root of order 2.
00:22 The square root cancels out the square. Isn't that neat?
00:29 Next, multiply the order of the first root by the order of the second root.
00:34 Now, apply this order as a root to our number.
00:38 Let's put this formula into practice with our exercise.
00:42 When you have a root of order C on number A to the power of B,
00:47 the result is A raised to the power of B divided by C.
00:52 Let's use this formula in our exercise again.
00:55 Break down 9 into 3 to the power of 2.
00:59 The square root cancels the square. See how it's done?
01:03 And that's how we find the solution! Great work!

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Simplify the left side of the equation, 81 \sqrt{\sqrt{81}} .
  • Step 2: Simplify the right side of the equation, x63 \sqrt[3]{\sqrt{x^6}} .
  • Step 3: Equate the simplified expressions and solve for x x .

Now, let's simplify each side:

Step 1: Simplify 81 \sqrt{\sqrt{81}} .

First, evaluate 81 \sqrt{81} , which is 9 9 , since 92=81 9^2 = 81 .
Then, evaluate 9 \sqrt{9} , which is 3 3 , since 32=9 3^2 = 9 .
So, 81=3 \sqrt{\sqrt{81}} = 3 .

Step 2: Simplify x63 \sqrt[3]{\sqrt{x^6}} .

Express x6 \sqrt{x^6} as (x6)1/2=x6/2=x3 (x^6)^{1/2} = x^{6/2} = x^3 .
Express x33 \sqrt[3]{x^3} as (x3)1/3=x3/3=x1=x (x^3)^{1/3} = x^{3/3} = x^1 = x .
So, x63=x \sqrt[3]{\sqrt{x^6}} = x .

Step 3: Set the simplified expressions equal.

We have simplified both sides of the equation to get 3=x 3 = x .
Therefore, the solution to the problem is x=3 x = 3 .

Hence, the correct answer is x=3 x = 3 .

Therefore, the correct choice is:

Choice 2: x=3 x = 3 .

Answer

x=3 x=3