Multiply or divide, as indicated. See Example 3. xz - xw + 2yz - ...

Question

Multiply or divide, as indicated. See Example 3. xz - xw + 2yz - 2yw 4z + 4w +xz +wx —————————— • —————————  z² - w² 16 - x²

Accepted Answer

Hey, everyone here we are asked to perform the indicated operations and simplify the following rational expression where we have K squared plus six K plus five all divided by K squared plus eight K plus seven divided by K squared plus K minus 20 all divided by K squared plus six K minus seven. Here we have four answer choice options, answer choice A K plus five all divided by K minus one answer BK minus one, all divided by K plus five. Answer CK minus one all divided by K minus four and answer DK plus five all divided by K minus four. So our first step before we perform any indicated operations is to factor the numerators and denominators of both fractions in our given expression. So starting with the numerator of our first fraction, it factors into the quantity of K plus one multiplied by the quantity of K plus five. And the denominator factors into the quantity of K plus one multiplied by the quantity of K plus seven. And now this is divided by the second fraction where the numerator factors into the quantity of K plus five multiplied by the quantity of K minus four. And now the denominator factors into the quantity of K plus seven multiplied by the quantity of K minus one. And now we can notice that in our first fraction, we can perform some immediate simplifications where here we see we have K plus one in both the numerator and the denominator. So both expressions cancel L. So now we just want to rewrite our current simplified expression, which will be the quantity of K plus five divided by the quantity of K plus seven. And then this is divided by our second fraction which remains the same for now with the numerator as the quantity of K plus five multiplied by the quantity of K minus four. And the denominator is still the quantity of K plus seven multiplied by the quantity of K minus one. And now we just want to recall that when we divide by a fraction, it is the same as multiplying by its reciprocal. So we want to rewrite our given expression using multiplication rather than division. So we have the quantity of K plus five divided by the quantity of K plus seven. And now we're multiplying by the reciprocal of our second fraction. So we are now multiplying where the numerator is now the quantity of K plus seven multiplied by the quantity of K minus one. And the denominator is now the quantity of K plus five multiplied by the quantity of K minus four. And now that we are multiplying, we can make further simplifications where we see that in the denominator of the first fraction, we have K plus seven and we have the same expression in the numerator of the second fraction. So both K plus sevens cancel out. And we can also notice that we have K plus five in the numerator of our first fraction as well as in the denominator of the second fraction. So both K plus five expressions cancel out as well. So now with all of these cancellations, we can rewrite our simplified expression, which is now just the quantity of K minus one divided by the quantity of K minus four. And with this, there are no further simplifications that we can make. So this is our final expression which leaves us with answer choice C where again, we have K minus one all divided by K minus four. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Multiply or divide, as indicated. See Example 3. xz - xw + 2yz - 2yw 4z + 4w +xz +wx —————————— • ————————— z² - w² 16 - x²

Key Concepts

Factoring Differences of Squares

Polynomial Simplification

Rational Expressions

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