Perform all indicated operations, and write each answer with posi...

Question

Perform all indicated operations, and write each answer with positive integer exponents. {x- 9y^-1}/{ (x-3y^-1)(x+3y^-1)}

Accepted Answer

Hello everyone. We're going to simplify the expression a minus 25 B. To the negative first Over a minus five B. The negative first times A plus five B to the negative first. First thing I want to do is clean up my negative exponents. I recall that in algebra if they're not grouped together it means that the negative exponents is only on the B. So my new numerator is a - over B. In my denominator I have some negative exponents there also but I think I can multiply this first and make it a little more clean. So I recognize this as the difference of squares which I recall means I don't end up with a middle term and I can square the first term giving myself a squared And square the last term resulting in 25 b to the negative second. And they should have a minus in between. Now that I have this all kind of put together, I'm going to work with my negative exponent in the denominator. Gonna rewrite my numerator without touching it for the time being. So that remains a -25 over B. In the denominator I have the same situation I did previously where the exponent is only on the B not on the 25. So my new denominator is a squared minus 25 over B squared. Next I'm going to try to make a common denominator for both my numerator and my denominator. So in my numerator the common denominator would be B. And this is achieved by multiplying A times B over B. So I have a B over B -25 over b. as my numerator in the denominator. My common denominator will be B squared. So I would do a squared B squared over B squared -25 over b squared. My next step, I'm going to make my numerator all one fraction and the denominator all one fraction. So when I do that I get a B -25 over B over a squared B squared minus 25 over B squared. Now that is one complicated looking fraction. Good news is if you recall that a fraction is the same as division. I can rewrite this as a division problem So I can do a B -25 over B. Divided by a squared B squared minus 25 over B squared. Now at least it's not all stacked on top of itself. It's a little easier to see remembering my fraction rules when I divide, I multiply by the reciprocal. So I get a B -25. My first fraction doesn't change. So over B. They changed my division to multiplication. And I flipped my second fraction to make the reciprocal. So I now have B squared over a squared B squared minus from here. I look to see is anything able to be canceled or should I factor something to reduce it. I recognize off the bat that I can cancel one of the bees with one of the b squared. So we get rid of that denominator. Then looking at a squared B squared minus 25 it does look like the difference of squares, but if I factor it, it's still not going to reduce with the numerator of the first fraction, so it might not be worth it. I'm gonna say let's clean this up and see if it matches any of our answer choices. So when I clean this up I get b times a B -25 over a squared b squared minus 25. And one last check. Does anything make sense to cancel? It does not. So this is my solution A times a B minus over a squared B squared minus 25 which is answer choice B. Have a nice day.

Perform all indicated operations, and write each answer with positive integer exponents. {x- 9y^-1}/{ (x-3y^-1)(x+3y^-1)}

Key Concepts

Rational Expressions

Exponent Rules

Factoring Polynomials

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