In Exercises 49–52, use the Binomial Theorem to expand each expre...

Question

In Exercises 49–52, use the Binomial Theorem to expand each expression and write the result in simplified form. (x^1/3 +x^-1/3)^3

Accepted Answer

Hey everyone here we are asked to use the binomial theorem to expand the given expression where we have the quantity of X to the 1/4 power plus X to the negative 1/4 power all raised to the power of four. Now, before we begin solving this problem here, we need to recall the binomial theorem which gives us a formula for expanding binomial and it tells us that for any positive integer end that the quantity of A plus B to the end power is equal to n zero times A. To the power of N plus n. One times A. To the power of n minus one times B plus and two times A to the power of n minus two times B squared plus and three times A to the power of n minus three times B. Cute plus. And then we we continue to expand our expression until we reach the term with the coefficient N. N. And times B to the power of N. So with this formula here we need to look at our given expression and start finding the values for these variables here. So we can begin with the a term from the formula and A in relation to our given expression is the left most term. So in this case our A. Is going to be X to the 1/4 power. And that leaves us with the other term which will be our B. So B is going to be X to the power of negative 1/4. And then lastly our end value is the power to which our by no meal is raised to. So in this case and is going to be four And now from here we can go ahead and start expanding. So once again we're expanding the expression X to the 1/4 power plus X to the negative 1/4 power all raised to the power of four. And so beginning with our first term again we replace N with four. So our coefficient here is 40 times A which is X to the 1/4 power to the power of N which is four. And then moving on to our next term we have plus four, one to times X to the 1/4 power. And now we raise this a value to the power of n minus one. So we raise X to the 1/4 power to the power of four minus one and then we multiply this by B. So we have times X to the negative 1/4 power and now we can move on to the next term here. So our coefficient is 42 and then we have times X to the 1/4 power to the power of four minus two times B which again is X to the negative 1/4 power. And we raise it to the power of to our next term we have the coefficient 43 times X to the 1/4 power raised to the power of four minus three times X to the negative 1/4 power raised to the power of three. And lastly we have our coefficient for the term which is 44. And we can see that this is the coefficient N N. In our formula. So it will be our last term here. And so we have X to the to the negative X to the negative 1/4 power raised to the power of four. And from here our expression is fully expanded so we can go ahead and start simplifying. So first with our coefficients here we need to recall that the coefficient of a binomial and our is equal to n factorial divided by the quantity of n minus R factorial times R factorial. And we need to know in this formula here that n is our upper value and our is our lower value. And so with this we can go ahead and start writing out our coefficients and also simplifying our X terms. So beginning with our first term, the coefficient becomes four factorial divided by the quantity of four minus zero factorial times zero factorial and simplifying our X term here, we multiply this coefficient by just x. Then for our next term the coefficient becomes four factorial divided by the quantity of four minus one factorial times one factorial. And this is the coefficient of the simplified X value which is X to the power of one half moving on to our next term. We have four factorial divided by the quantity of four minus two factorial times two factorial and this is the coefficient of this simplified X term here which simplifies to just one next we can move on. So we have four factorial divided by the quantity of four minus three factorial times three factorial and simplifying the X terms. We multiply this coefficient by X to the power of negative one half. And with our last term here the coefficient is four factorial divided by the quantity of four minus four factorial times four factorial and this is the coefficient of X to the power of negative one. So now we can continue to simplify our coefficients here. So our next simplification can just be performing the subtractions in the denominators of each coefficient. So beginning with our first term we have four factorial divided by four minus zero, gives us four factorial times zero factorial and this is the coefficient of x plus the next term we have four for factorial divided by four minus one, gives us three factorial times one factorial and this is the coefficient of X. To the power of one half plus our next term we have four factorial divided by four minus two, gives us two factorial times two factorial, all times one plus our next term four factorial divided by four minus three gives us one factorial times three factorial and our x term here is X to the power of negative one half. And with our last term we have four factorial divided by four minus four gives us zero factorial times four factorial and the X term here is X to the power of nega one. And now we can continue to simplify here. So again looking at our coefficients starting with our first term, we see that we have a four factorial in the numerator and the denominator so they cancel. And we also need to recall that zero factorial is equal to one. So from here we can see that we have a coefficient of one. So we have one x for our first term for our next term we have this three factorial in the denominator so we can expand our numerator until we also have three factorial. So doing this in our numerator we have four times three factorial, all divided by three factorial times one factorial and this is the coefficient of X to the power of one half. Moving on we have a two factorial in our denominator of our next coefficient here. And so we can again expand our numerator until we also have a two factorial in the numerator. So doing this, we have four times three times two factorial. All divided by two factorial and we have times another two factorial but we can go ahead and expand it to two times one and this is the coefficient of just one. Moving on to our next term. We again have this three factorial in the denominator. So we can again expand our numerator we get four times three factorial all divided by one factorial times three factorial. And this is the coefficient of X to the power of negative one half. And lastly with our last term we see that the coefficient is the same as our first turn again, the four factorial is cancel out. And since zero factorial is just one, we are left with the coefficient of 14 X to the power of negative one. And now we can just continue to simplify here. So our first term, we can exclude the one in front since it is implied by the X being alone and then moving on to our second term here we see that the three factorial is canceled out. So we have four divided by one factorial, which is just one. So we are left with four Times X to the power of one half. Next for our next term. We see that the two factorial is canceled out and then we are left with 12 divided by two giving us six times one. So we just have six for this next term. Moving on we have four times three we see that. Sorry, we see the three factorial is canceled out again. So we have four divided by one factorial which simplifies to just four. So we have four times X to the power of negative one half for our next term here. And lastly we you can just rewrite the last term without the one coefficient. So our last term is X to the power of negative one. And so from here, the last thing to do is to simplify this using the rule of exponents where we have that A to the power of negative M. Is equal to one divided by A. To the power of positive M. So again we can use this to rewrite our expansion here and rewrite its simplified form. So we see we are again expanding the quantity of X to the 1/4 power plus X to the negative 1/4 power raised to the power of four. And now our final simplified expansion is X plus four times X to the power of one half plus six plus. And now rewriting our next turn, we have four divided by X to the power of one half plus and rewriting our last term. We have one divided by X. And with this we have fully expanded our expression here and simplified the results and we are left with answer choice B. Thanks for watching. I hope you found this video helpful. And I'll see you again next time.

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In Exercises 49–52, use the Binomial Theorem to expand each expression and write the result in simplified form. (x^1/3 +x^-1/3)^3