entire radicals to mixed radicals

Question

√180x^4

Answer 1

First, let's write 180 as a product of its prime factors:
180 = 2 * 2 * 3 * 3 * 5 = 2^2 * 3^2 * 5

Next, let's simplify the square root of each prime factor:
√2^2 = 2
√3^2 = 3
√5 = 5

Therefore, we can rewrite the original expression as:
√180x^4 = √(2^2 * 3^2 * 5 * x^4)

Using the product property of square roots, we can split the radical:
√(2^2 * 3^2 * 5 * x^4) = √(2^2) * √(3^2) * √(5 * x^4) = 2 * 3 * √(5 * x^4)

Simplifying further, we have:
2 * 3 * √(5 * x^4) = 6√(5x^4)

So, √180x^4 is equivalent to 6√(5x^4) as a mixed radical.

Answer 2

why stop there?
√180x^4 is equivalent to 6x^2 √5

Answer 3

You are correct, I apologize for stopping the simplification process prematurely. Let's continue simplifying √180x^4.

We have:

√180x^4 = √(2^2 * 3^2 * 5 * x^4)

Using the product property of square roots, we can split the radical:

√(2^2 * 3^2 * 5 * x^4) = √(2^2) * √(3^2) * √(5 * x^4) = 2 * 3 * √(5 * x^4)

Simplifying further, we have:

2 * 3 * √(5 * x^4) = 6 * √(5 * x^4)

Since x^4 is equal to (x^2)^2, we can rewrite the expression as:

6 * √(5 * (x^2)^2)

Since the square root of a square term cancels out, we have:

6 * √(5) * x^2

Therefore, √180x^4 is equivalent to 6x^2√5 as a mixed radical. Thank you for catching that mistake!