In Exercises 93–102, factor and simplify each algebraic expres...

Question

In Exercises 93–102, factor and simplify each algebraic expression. x^3/4−x^1/4

Accepted Answer

Hey everyone in this problem we are asked to factor and simplify the following algebraic expression which involves fractional exponents. Okay we're giving the expression x minus X to the exponent one third. Okay so our expression X -X to the exponent 1/3. Now when we're factoring we want to take out the lowest common factor first. And what we'll see is that lowest common factor is going to be X to the exponent one third. In our first term here, X we have X to the exponent one. In this second term X to the exponent one third. One third is less than one. So that's the factor. We're gonna take it. So we take X to the exponent 1/3. What's going to be left? Well when we remove this factor it's like dividing. Okay, so we're going to take X divided by X to the exponent one third. Right? And this is gonna give us X to the exponent one minus a third. Okay, when we divide with the same base, the exponents subtract now. The second term X to the exponent one third. We divide by X to the exponent one third and we just are left with one. Okay, we're taking that whole factor out. Alright, so let's simplify this first X to the exponent one third times X to the one minus one third. This is going to be X to the two thirds minus one. And you can always check at this point if you go and and expand. Okay so we know that the X to the one third is going to apply to both terms. We get X to the one third times X to the two thirds. When we multiply things that have the same base, we add the exponents one third plus two thirds gives us one. Okay so we're back to that original equation so we can always double check by expanding. Alright now let's look at this. Okay, we're gonna rewrite what's inside the brackets to make this look maybe a little more familiar. Okay we can write X to the exponent two thirds as X to the exponent. One third squared When we raise to the power we multiply the exponents. So this is X to the 2/3 and one. Well we can write one as one squared. So now we have something squared minus something else squared. Okay, so this is going to be a difference of squares. Okay and recall if we have a difference of squares a squared minus B squared, we can write this as a plus B Times A. -7. Alright, so if we apply that to our expression we get X to the exponent one third times well in our case A is X to the one third, so we get X to the one third And B is one. So we get plus one. Okay and then the second factor is X to the one third minus one. Alright, so now we've completely factored this. Okay at first we pulled out the lowest common factor between our two terms and then we use the difference of squares to factor that remaining term. Okay? And we are left with answer C. Okay? We get X to the exponent one third times X. To the one third plus one times X. To the one third minus one. Okay? And the notation here is a little different. Can we get the cube root? And the cube root recall is the exact same thing as writing X to the exponent 1/3. Thanks everyone for watching. See you in the next video.

In Exercises 93–102, factor and simplify each algebraic expression. x^3/4−x^1/4

Key Concepts

Factoring Algebraic Expressions

Exponents and Radicals

Common Factor Extraction

Watch next

In Exercises 93–102, factor and simplify each algebraic expression. x3/4−x1/4

Key Concepts

Factoring Algebraic Expressions

Exponents and Radicals

Common Factor Extraction

Watch next

In Exercises 93–102, factor and simplify each algebraic expression. x^3/4−x^1/4