I need help with this last math problem i have: Using the properties for radicals to simplify, assuming all variables represent positive real numbers.

square root 12x^3/5

First you have to realize that:

square root 12x^3/5
and
(12x^3/5)^1/2

are equal! :-)

Taking the square root of a value is the same as taking the value to the 1/2 power

Use this rearrangement to simplify:

(12x^3/5)^1/2
Multipy the exponents...

12x^3/10

I'm pretty sure that's as far as you can go with the simplifying

Good luck! :-)

The answer depends upon whether you mean

[sqrt (12 x)]^3/5
or (sqrt 12) x^3/5,
or sqrt (12 x^3/5)
The way you wrote it is ambiguous. You need to use parentheses to clarify what you are taking the square root OF.

If you meant the last of these, then it is equal to (sqrt 12) * x^0.3

To simplify the given expression, we can begin by splitting the radicand (the expression inside the square root) into individual factors.

Starting with the numerator, the factors of 12 are 2 and 6.

Next, we examine the exponent in the denominator. Since the exponent is 5, we can rewrite x^3 as x^2 * x.

Now, let's simplify the expression:

√(12x^3/5) = √(2 * 6 * x^2 * x / 5)

We can apply the properties of radicals to simplify further. The square root of each factor can be separated and placed outside the square root symbol:

= (√2 * √6 * √x^2 * √x) / √5

Since √x^2 simplifies to x (assuming x is a positive real number), the expression becomes:

= (√2 * √6 * x * √x) / √5

Finally, we can multiply the square roots together and put them back inside the square root symbol:

= (√(2 * 6 * x^2 * x)) / √5

Simplifying further:

= (√(12x^3)) / √5

Therefore, the simplified expression for the square root of 12x^3/5 is (√(12x^3)) / √5.