In Exercises 83–90, perform the indicated operations. Simplify th...

Question

In Exercises 83–90, perform the indicated operations. Simplify the result, if possible. (y^−1−(y+2)^−1)/2

Accepted Answer

Hey, everyone. Here, we are asked to simplify the result after executing stated operations. And we are given the expression the quantity of four plus three X divided by the quantity of two minus X times the quantity of one plus six divided by the quantity of X minus eight. Here we have four answer choice options. Answer a one answer B X answer C XX minus two and answer D X minus eight. So to begin this problem, we first want to combine our whole numbers in each set of brackets into the numerator of the fractions. So beginning with our first set of brackets, we see we have the whole number of four. Well, four is equivalent to four divided by one. And so we want to take our four divided by one and multiply the numerator and the denominator by the denominator of the fraction. So here we multiply the numerator and denominator by two minus X. So doing this, we have four times the quantity of two minus X all divided by one times the quantity of two minus X. Then we have our fraction here plus three X divided by two minus X. And now we can repeat this process with the other set of brackets. Again, we take our whole number here, which is one and we know one is the same as one divided by one. And here we need to multiply the numerator and denominator of one by the denominator of the fraction, which is X minus eight. So combining these fractions by first finding a common denominator, we have one times the quantity of X minus eight all divided by one times the quantity of X minus eight. Then we still have our other fraction here. So we have plus six divided by the quantity of X minus eight. So now we just want to distribute in our whole numbers for both set of brackets here. So beginning again with our first set of brackets, we can distribute the four in the numerator as well as the one in the denominator. So we have eight minus four X all divided by two minus X. And then we have plus three X divided by the quantity of two minus X. And now, for the second set of brackets, we again distribute the numerator whole number and the denominator whole number, which is both one. So we have X minus eight all divided by X minus eight. And then we still have the other fraction which is plus six divided by the quantity of X minus eight. So now we can see that for both set of brackets here, both fractions have a common denominator. So the next step is to just combine our numerators. So beginning again with our first set of brackets in the numerator, we now have eight minus four, X plus three X all divided by two minus X. And in the second set of brackets here, we have X minus eight plus six all divided by the quantity of X minus eight. And so now we just want to combine like terms that are in our numerators here. So once again, beginning with our first set of brackets, combining the X terms in the numerator, we have eight minus X all divided by two minus X. And now in our second set of brackets, we have the simplified form of X minus two, all divided by X minus eight. So now looking at our first set of brackets, we can notice that in the numerator and the denominator, we can factor out a negative one. So factoring out a negative one in the numerator and the denominator, we have negative one times the quantity of X minus eight. And this is all divided by negative one times the quantity of X minus two. And now we will keep the second set of brackets the same here. So we have X minus two, all divided by the quantity of X minus eight. So now by factoring out this negative one in the numerator and the denominator, we see that by cross multiplying, we can cancel out the expressions here we can cancel out our X minus eight since it appears in the numerator of our first fraction in the denominator of our second fraction. And now in the denominator of our fir first fraction, we have X minus two which cancels out with the numerator of the second fraction. So now just simplifying, we see here that we are left with negative one divided by negative one. And simplifying, we see that this is just one. So here we see after simplifying our given expression that we are left with answer choice. A where the simplification is just one. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 83–90, perform the indicated operations. Simplify the result, if possible. (y^−1−(y+2)^−1)/2

Key Concepts

Negative Exponents

Combining Fractions

Simplifying Expressions

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