Can someone please help me simplify root36x^3? I am not sure how to use the root symbol on here but the problem is the root symbol with a 36x^3 inside of it. Thank you!

Question

anonymous · Answer

do the 36^3 first then take the square root

anonymous · Answer

so 36^3=46,656. Now take the square root which is 216

anonymous · Answer

Somehow the people in my class are coming up with 6x and 6xrootx for answers, and I don't understand how to get that.

anonymous · Answer

What do I do next after getting the 216?

anonymous · Answer

thats the amswer

anonymous · Answer

what are you studying

anonymous · Answer

Algebra right now. but I don't understand how people are getting the 6x for an answer then?

ciarán95 · Answer

I presume this is what you are looking to simplify:
$$\sqrt{36x ^{3}}$$

Taking the square root of something is the same as raising that same thing to the power of 1/2 (in the same way that the cubed root is the same as raising to the power of 1/3, and so on...). So, we can write the above as:
$$(36x ^{3})^{\frac{ 1 }{ 2 }}$$

One of the rules of indices states that, for any two real numbers 'a' and 'b':
$$(x ^{a})^{b} = x ^{ab}$$
and that, for example:
$$(3x ^{a})^{b} = 3^{b}x ^{ab}$$
So, you can multiply the power on the 36 (essentially just 1) and the power on the x (3) each by 1/2 to get the new powers.

So, we get:
$$36^{\frac{ 1 }{ 2 }}x ^{\frac{ 3 }{2 }}$$, and you can complete by simplifying the 36^1/2 part to get the answer. Hope that helps you out! :)

anonymous · Answer

Yes. that is exactly what I am trying to solve! So where does the 3/2 come from and how would I simplify the 36^1/2 part to get the answer?? Yes you are helping me out greatly!

ciarán95 · Answer

Let's look and see what we did when we multiplied it powers on 36 and x by the overall power we were raising the expression to, 1/2.

As I stated above, if we have an expression which contains 'something' raised to a power (like x^2) and we raise this entire expression to a power (like (x^2)^2), then we multiply the two powers to give us a new power, to which this 'something' will now be raised to (e.g. x^4).

$$(36^{1}x ^{3})^{\frac{ 1 }{ 2 }}$$
We're just using the two rules I stated above in my previous post in order to simplify.

In simplifying the '36' part of the expression, we multiply the power on '36', which is just 1, by the power we are raising the whole expression to, and this will become the new power on the '36'. So, as 1 x 1/2 = 1/2, then we are left with 36^1/2.

This is the exact same thing were doing when we are considering the 'x' part of the expression. The 'x' is raised to the power of 3, so 3 x 1/2 = 3/2, and this becomes the new power on the 'x'. This is how we are left with:

$$36^{\frac{ 1 }{ 2 }}x ^{\frac{ 3 }{ 2 }}$$

We can't really simplify the 'x' part of the expression much further, so we're done with that part. However, you can work out 36^1/2, or what we can also call 'the square root of 36', on a calculator pretty simply. When you work this out this should give you the final simplified expression!
$$36^{\frac{ 1 }{ 2 }} = \sqrt{36} = ?$$

anonymous · Answer

The square root of 36 is 6. So is 6x the answer then? Or would it just be 6? I am still confused kind of but understanding a little better anyway

ciarán95 · Answer

Say we wanted to simplify the following:
$$(2^{2})x ^{4}$$

What we'd do is simply work out 2^2, which is 4, and substitute it back into the expression. So:
$$(2^{2})x ^{4} = 4x ^{4}$$
We can't do anything to the x^4 part which would simplify the expression further. If we were to try an do anything to it, we would either be working backwards and getting an more complicated expression, or we would end up with the same answer.

As you've correctly said, 36^(1/2) is indeed 6. So, all we are doing is making 36^1/2 simpler (i.e we've turned it into 6, and we can't make that any simpler!) and putting '6' back into the expression in place of it. So, we get:
$$36^{\frac{ 1 }{ 2 }}x ^{\frac{ 3 }{ 2 }} = 6x ^{\frac{ 3 }{ 2 }}$$

Hopefully from doing out that last part you understand the idea a bit better now. We're just making part of the expression easier to read, and not touching the part of it which we can't simplify anymore! :)