entire radicals to mixed radicals

Question

√96x^3

Answer 1

To convert √96x^3 from entire radicals to mixed radicals, we first need to find the largest perfect square factor of 96x^3.

Since 96 is not a perfect square, we can find its prime factorization: 96 = 2^5 * 3^1.

To find the largest perfect square factor of 96x^3, we can look at the exponents of the prime factors. The largest perfect square factor is 2^4 = 16.

We can then write √96x^3 as √(16 * 6 * x^2 * x).

Using the property of radicals, we can split the square root and simplify:

√(16 * 6 * x^2 * x) = √(16) * √(6) * √(x^2) * √(x) = 4√(6x^2) * √(x).

So, √96x^3 can be written as 4√(6x^2) * √(x), which is a mixed radical.

Answer 2

why stop there?

√96x^3 = √(16x^2 * 6x) = 4x√(6x)

Answer 3

You're right, I apologize for stopping prematurely. We can simplify further by taking out any perfect square factors from under the square root.

√(16x^2 * 6x) = √(16x^2) * √(6x) = 4x * √(6x).

So, √96x^3 can be written as 4x * √(6x), which is a simpler mixed radical. Thank you for pointing that out!