In Exercises 55–78, use properties of rational exponents to simpl...

Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers.

(x^½y^-⅗)^½

Accepted Answer

Hey, everyone in this problem, we're asked to simplify the following expression by using the properties of irrational exponents. We're told to assume that A is greater than zero and B is also greater than zero. The expression given is A to the exponent 1 4th multiplied by B to the exponent negative 5/ all to the exponent 1/4. We're given four answer choices. Option A A to the exponent one divided by 16 multiplied by B to the exponent five divided by 28. Option B A to the exponent one half divided by B to the exponent 13 divided by 28. Option C A to the exponent 1 divided by B to the Exponent 5 28. And option D A to the exponent one half multiplied by B to the exponent 13 divided by 28. We're gonna start by rewriting what we were given. So in brackets, we have A to the exponent 1 4th multiplied by B to the exponent -5/7. And all of this is raised to the exponent 1/4. Now we have two things multiplied together in a term and our B term that are raised to an exponent, that exponent of 1/4 has to apply to both of these terms. OK. We have to distribute this exponent. How can we do that? Well, we call, if we have two things multiplied together, say X multiplied by Y to the exponent Z, we can write this as X to the exponent Z multiplied by Y to the exponent Z OK. That exponent applies to both of those terms. So we can separate it, apply it to each term and multiply the result together applying this to our expression, we can write a to the exponent 1 4th to the exponent, 1 4th multiplied by B to the exponent negative five sevens to the exponent 1 4th. So now this looks a little bit more like exponents. We're used to dealing with here where we have a single base, but now we have an exponent raised to an exponent. How do we deal with that? Well, we're called the power rule. We have X to the exponent M to the exponent N, you can write this as X to the exponent M multiplied by N. So when we have something raced to an exponent, let's raise to another exponent. We can simplify this into a single exponent by multiplying the two together applying this. In our case, we have a to the exponent 1/4 to the exponent, 1/4 we multiply the two exponents together 1/4 multiplied by 1/4 gives us 1/16 And so we get A to the exponent 1/16. Now, for the B term, we're gonna do the same thing B to the exponent negative 5/ to the exponent 1/4 multiply the exponents together negative 5/7 multiplied by 1/4 gives us B to the exponent negative five divided by 28. Now we've simplified quite a bit. You'll notice that the answer choices we were given all have positive exponents only and we have a negative exponent on our B term. So how can we rewrite this to have positive exponents? Only? We recall if we have a negative exponent, say X to the exponent negative M, we can write this as one divided by X to the exponent positive F. So if we move the base and the exponent to the denominator, we can switch that exponent to become positive. Applying this in our problem. We have a to the exponent one divided by 16. OK. We're not simplifying that any further multiplied by one divided by B to the exponent five divided by 28. OK. So we've moved the base and the exponent to the denominator and made it positive. And now we can just simplify the way we've written this. Our expression becomes A to the exponent 1/ divided by B to the exponent 5 28. And that is our simplified ex expression. We have only positive exponents. That's what we were looking for. OK. So we found the correct answer is A to the exponent 1/ divided by B to the exponent 5 20 eights which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. (x^½y^-⅗)^½

Key Concepts

Rational Exponents

Properties of Exponents

Simplifying Expressions

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