In Exercises 39–64, rationalize each denominator.3xy²-----------⁵...

Question

In Exercises 39–64, rationalize each denominator.3xy²-----------⁵√8xy³

Accepted Answer

Hello, everyone. We are going to eliminate the radical in the denominator by rationalizing the expression we are given is five A squared B over the fifth route of A cubed B. Our answer choices are A five B multiplied by the fifth root of nine A squared B raised to the fourth power over three B five A multiplied by the fifth root of nine A squared B raise the fourth power over three, see three B multiplied by the fifth route of nine A squared B raise the fourth power over five A D three A multiplied by the fifth root of nine A squared B raised the fourth power over five B. I'm gonna rewrite my fraction. So five A squared B Over the 5th route of 27 A cubed B recall that to rationalize a denominator, we want to be able to cancel out the fifth route on every value that is under that radical. So we need to first come up with what would make it. So the fifth route would cancel off each base. So to figure that out, I'm gonna rewrite my numerator because that part is not gonna change just yet. So I have currently the 5th route and I'm Gonna Rewrite 27. I recognize that 27 is three cubed, knowing what exponent goes on that will help us. And then we have a cubed and it's B to the first. The reason this helps us is because what we want to multiply both the top and bottom of our fraction by Is going to be the 5th route of all the values that when we multiply by our bases in our original will give us an exponent of five. What I mean is when you recall that we add our exponents, when we multiply to get three, raise the third power to be three, raise the fifth power. It needs to be multiplied by three squared To get a cubed to be a raised to the 5th power, it needs to be multiplied by a squared And to get B raised the first power to be B raised the fifth power, it needs to be multiplied by B raised to the 4th power. So the factor we're gonna multiply everything by is the 5th route of three squared A squared B raise the fourth power. So this is gonna be the numerator and the denominator that I'm gonna multiply by. So it is now both the top and the bottom of the fraction. I'm going to multiply my original fraction by. So in doing so for the numerator, I'm going to just write them directly next to each other for now. So five a square b Multiplied by the 5th route of three squared A squared B race the fourth power if there's a way to simplify that after I will. But at this step, I'm just writing them directly next to each other. When I multiply the denominators, I'm gonna keep it under the 5th rou for this step, I will simplify it in the next. So it's still the 5th route. A cubed multiplied by or sorry, three cubed multiplied by three squared Is three, raise the 5th power A cubed multiplied by a squared will be a raise the 5th power B raise the first power multiplied by B raise the 4th power is B raise the 5th power. So it worked out how we wanted it to, we have all of our exponents in the radical that's in our denominator raised to the fifth power. So we're gonna to simplify this N reader is gonna stay the same for now. So five A squared B raised the fifth power multiplied by oh sorry, not B raise the fifth power B multiplied by the fifth route of three squared A squared B to the fourth In my denominator, the fifth route and the five exponents will cancel. So this will leave me with three A B as my denominator. Now, I wanna see if there's anything else I can simplify. I wanna reduce my numerator with my denominator. If at all possible, my numerator has an A squared. My denominator has an A so I'm gonna leave the five and I'm gonna reduce A squared to A and that'll cancel out the A in my denominator. They also both have A B which I can also cancel out. So that doesn't need to be there anymore. Three remains in my denominator. Simplifying anything in the Num Reader that's under the 5th route. I'm not gonna be able to take the fifth route of any of it, but I can do that. Three squared is nine A squared and b to the 4th will stay put. So if this looks as simplified as it can get, we have five a multiplied by the fifth route of nine A squared B raised the fourth over three and that matches answer choice B have a nice day.

In Exercises 39–64, rationalize each denominator.3xy²-----------⁵√8xy³

Key Concepts

Rationalizing the Denominator

Radical Expressions

Properties of Exponents

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