Help on simplifying radicals with variables?

Question

I still don't understand how to simplify radicals. In the example √36x^2y^12, it equals 6√x^2y^12, which equals 6xy^6. I understand the 6 and the x, but I don't understna how you get y^6.
When you find the square roots of variables to powers, don't you divide the even numbered power by two to take the variable out of the radical?
And how do you do odd powered powers? For example, when you have √8x^3, don't you get 2√2x^2x, which equals 2x√2x ?

Answer 1

a) √36. b)√x². c)√y¹²
Factoring separately:
36| 2
18|2
09|3
03|3
1
a) √36 = √2². √3² =
Simplifying the exponents by radicals =
2.3 = 6
b)√x² = x
c)√y¹² = y^6
Solving together:
a).b).c) = 6xy^6
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Odd powered powers:
√8x³ =
8|2
4|2
2|2
1
√8 = 2.2.2 = 2³ = √2.√2² = 2√2
√x³ = x.x.x = √x². √x = x√x
Together: 2x√2x
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Answer 2

Note that

sqrt(36[x^2][y^12]) is the same as

(36[x^2][y^12])^(1/2)

By the exponential property that

(abc)^m = (a^m)(b^m)(c^m), our expression above equals

(36)^(1/2) (x^2)^(1/2) (y^12)^(1/2)

By the exponential property (a^b)^c = a^(bc) {i.e. whenever we have a power to a power, we can multiply the powers}, we get

(6) (x) (y^6)

{Notice how I multiplied 1/2 with each of the exponents, which is the same as taking half of the value.}

Therefore, our final answer is

6xy^6

If you have sqrt(8x^3), then

sqrt(8x^3) is the same as sqrt( 4 * x^2 * 2 * x)

For every perfect square you pull out of the square root (x^2 is a perfect square), it comes out as a singular version of itself. Therefore, the 4 would come out of the square root as 2, and the x^2 would come out of the square root as x, and you'd be left with

2x sqrt(2x)

Answer 3

in the 1st situation, all you're able to do is discover the sq. root of 36 and x^2 on the grounds that they the two are appropriate squares. on the grounds that 12 isn't a appropriate sq., divide it by using 2. (there is not any selection in the unconventional image, so that's understood to be 2). y^12 divided by using 2 is y^6.... Your selection 2 situation is right.

Answer 4

y^12 = y*y*y*y*y*y*y*y*y*y*y*y

= (y*y*y*y*y*y) * (y*y*y*y*y*y)

so sqrt y^12 is y*y*y*y*y*y which is y^6

Yes u do divide the even power index by 2 to take a number of a variable out of the sqrt sign.

you can do odd powers like what you've said. however whether it makes it any simpler is up to the question.

Answer 5

Look at it this way. George Bush is a radical. Ellen DeGeneres is a variable...