Solve the Square Root Equation: x^6 = √16 · √25

Question

Solve the following exercise:

x6=16⋅25 \sqrt{x^6}=\sqrt{\sqrt{16}}\cdot\sqrt{25}

Video Solution

Solution Steps

00:10 Let's solve this problem together.
00:14 A regular root is usually of the order two, like a square root.
00:20 We can express sixteen as four squared.
00:24 And twenty-five as five squared.
00:28 When we take the C-th root of A to the power of B,
00:31 It's like saying A to the power of B divided by C.
00:36 Let's use this formula on our exercise.
00:40 Notice that the root cancels out the square.
00:44 Now, calculate the power quotient.
00:50 We can also express four as two squared.
00:54 And again, the root cancels the square.
00:58 That's how we solve this problem, great job!

Step-by-Step Solution

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

  • Step 1: Simplify 16 \sqrt{\sqrt{16}} .
  • Step 2: Simplify 25 \sqrt{25} .
  • Step 3: Calculate the right-hand side.
  • Step 4: Solve x6= \sqrt{x^6} = computed right-hand side.

Now, let's work through each step:
Step 1: Simplify 16 \sqrt{\sqrt{16}} .
We know that 16=4 \sqrt{16} = 4 , and 4=2 \sqrt{4} = 2 . So, 16=2 \sqrt{\sqrt{16}} = 2 .

Step 2: Simplify 25 \sqrt{25} .
We know that 25=5 \sqrt{25} = 5 .

Step 3: Calculate the entire right-hand side.
We have 16⋅25=2⋅5=10 \sqrt{\sqrt{16}} \cdot \sqrt{25} = 2 \cdot 5 = 10 .

Step 4: Solve x6=10 \sqrt{x^6} = 10 .
Rewrite the left side as (x6)1/2=x6/2=x3 (x^6)^{1/2} = x^{6/2} = x^3 .
Thus, x3=10 x^3 = 10 .

Therefore, the solution to the problem is x3=10 x^3 = 10 .

Answer

x3=10 x^3=10

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Square Root Rules Combining Powers and Roots Square Roots Combining root laws

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