simplify express with positive exponents 6x^-3y^4/15x^2y^-2?

Question

Simplify...express with positive exponents

six x to the negatative third power y to the fourth power over fifteen x to the second power y to the negatative second power.

6x^-3y^4/15x^2y^-2

Please show me how you do it..not just give an answer.

Answer 1

Remember the rule for division of exponents of the same base:

(X^m)/(X^n) = X^(m-n)

So, lets start by adding a couple parenthesis to make it look a bit more clear:

. 6x^-3y^4/15x^2y^-2
=[6(x^-3)(y^4)] / [15(x^2)(y^-2)]

You have two types of bases here, x and y, and both are present in both the numerator and the denominator. That means you can do that subtraction of exponents rule in each one:

. [6(x^-3)(y^4)] / [15(x^2)(y^-2)]
= (6/15) * [(x^-3)(y^4)] * [(x^2)(y^-2)]
= (6/15) * [(x^-3)/(x^2)] * [(y^4)*(y^-2)]
= (2/5) * x^(-3-2) * y^(4-(-2))
= (2/5) * x^(-5) * y^(6)
= 2(x^-5)(y^6) / 5

Now just remember the fact that x^-n = 1 / x^n, and we can convert that one negative exponent (x^-5) to a positive one by moving it to the denominator (1/x^5):

= 2y^6 / 5x^5

---------------

You could also approach this whole problem in a different order, namely by remembering the fact that x^-n = 1/ x^n and expressing this fact by showing both your numerator and your denominator as fractions in positive exponents.

So, starting with:

6x^-3y^4 / 15x^2y^-2,

express the numerator and denominator both as fractions in positive exponents of x and y:

(6y^4/x^3) / (15x^2/y^2)

Remember then that to divide fractions, you multiply by the inverse so:
. (6y^4/x^3) / (15x^2/y^2)
= (6y^4/x^3) * (y^2/15x^2)

Now you are just multiplying fractions, so multiply the numerators together and multiply the denominators together:

= 6y^4y^2 / 15 x^3x^2

and remember that exponents of the same base add together when multiplied, so you get:

= 6y^6 / 15x^5

and finally simplify the numbers to lowest terms:

= 2y^6 / 5x^5

------------

Another way you could do this is to express both the previous fact and its inverse (the fact that 1/x^-n = x^n) as the first step simultaneously:

. 6x^-3y^4 / 15x^2y^-2

move the inverse of x^-3 to the denominator and
move the inverse of y^-2 to the numerator:

= 6y^4y^2 / 15x^2x^3

remember that multiplying exponents of the same base is done by adding the exponents of that base:

= 6y^(4+2) / 15x^(2*3)
= 6y^6 / 15x^5

and as before, simplify the numbers to lowest terms:

= 2y^6 / 5x^5

= = = = = = = = =

Sorry for such a long answer, but I wanted to show that there are many different ways of approaching it. Once you have memorized the rules for operations with exponents, you can break such a problem down any way that makes the most sense to you. The "right" way will eventually depend on where you are trying to go with the answer - so for example eventually you will be dealing with "imaginary numbers" (numbers involving the imaginary value "square root of negative one"), and so you will choose the approach that isolates that imaginary value as cleanly as possible.

It looks to me like the other answers here are making the mistake of inverting the multipliers of the base of the exponents along with the base and exponent itself:

5x^-2 = 5 * x^(-2) = 5 * 1/(x^2), not
5x^-2 = (5x)^(-2) = 1/(5x)^2

You can check this by assigning x a value -- let's use 10:

10^(-2) = .01 and 10^2 = 100, so

5x^-2 = 5 * 10^(-2) = 5 * .01 = .05
5 * 1/(x^2) = 5 * 1/100) = 5 * .01 = .05

versus

(5x)^(-2) = 50^(-2) = 1/(50^2) = 1/2500 = .0004

But, maybe it's just me who is tired (it is 1:20 AM here), so double check both my work and theirs.

Answer 2

Hmm. All numbers with negative exponents are equivalent to one over that number raised to a positive exponent. That is:


1
6x^-3 = ------
6x^3


And,


1
y^-2 = ------
y^2



Therefore, your equation:



6x^-3y^4
-----------
15x^2y^-2



Which can also be written as:


6x^-3 1 y^4 1
------ * ------- * ------ * ------
1 15x^2 1 y^-2


May be re-written as:


1 1 y^4 y^2
----- * ------ * ----- * -----
6x^3 15x^2 1 1



Simplifying:


y^4 * y^2 y6
---------------- = ----------
6x^3 * 15x^2 90x^5



This may appear all jumbled up once this is posted but the final answer is: y raised to the sixth power over ninety x raised to the fifth power.

Answer 3

6x^-3*y^4/15x^2*y^-2 =
y^4*y^2/90*x^3*x^2 =
y^6/90x^5=
(y(y/x)^5)/90