entire radicals to mixed radicals
√180x^4
why stop there?
√180x^4 is equivalent to 6x^2 √5
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!
To simplify √180x^4, we can break it down into its prime factors and then group them into perfect square factors and remaining factors.
Step 1: Prime factorize 180
180 can be broken down into prime factors as follows:
180 = 2 × 2 × 3 × 3 × 5
Step 2: Group perfect square factors and remaining factors
√180 x^4 can be written as:
√(2 × 2 × 3 × 3 × 5) x^2 x^2
Simplifying each group separately:
√(2 × 2) × √(3 × 3) × √5 × x^2 × x^2
In square roots, a perfect square can be taken out from under the root sign. Since we have two 2's, two 3's, and one x^2 inside the square root, we can take one 2, one 3, and one x out of the square root.
So, the simplified form is:
2 × 3 × √(2 × 5) × x × x
This can be further simplified as:
6x√(10)
Therefore, the simplified form of √180x^4 is 6x√10.
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.