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.

(25x⁴y⁶)^½

Accepted Answer

Hey, everyone in this problem, we're asked to simplify the following expression by using the properties of rational exponents. We're told to assume that A is greater than zero and B is greater than zero. The expression we're given is 216 A to the exponent nine B to the exponent 15 all to the exponent one third, we have four answer choices. Option A six A cubed B to the exponent five. Option B 72 A cubed B to the exponent five. Option C six A to the exponent divided by three B to the exponent 46 divided by three and option D six A to the exponent 26 divided by three B to the exponent 44 divided by three. We're gonna start by rewriting what we were given. So we have in brackets 216 a to the exponent 9 b to the exponent 15. And all of this is raised to the exponent 1/3. Now we have some exponents inside of our brackets. We have three terms multiplied together that are raised to an exponent. So how do we deal with applying that exponent? Well, that exponent has to apply to all three of those terms. So we need to distribute the exponent of 1/3 to each of these terms. So we call, if we have two things multiplied together, say X multiplied by Y to the exponent M, we can write this as X to the exponent M multiplied by Y to the exponent M. So we apply that exponent to each term individually and multiply the result. Now applying this to our problem, we have three things multiplied together. So we're gonna have 216 to the exponent. 1 3rd multiplied by A to the exponent nine to the exponent 1 3rd multiplied by B to the exponent 15 to the exponent 1 30. So now we've broken this up into three little pieces that we can deal with individually multiplying them together to get our final result. Now, starting with the first term, we have 216 to the exponent 1/3, recall that an exponent one third is equivalent to taking the third route. So what we're looking for is we're looking for a number that when we raise it to the exponent three gives us 216. Well, that number is six, OK, six cubed is equal to 216. So 216 to the exponent one third is going to give us six. Now moving on to our terms with A and B, we have an exponent raised to an exponent. How do we deal with that? We recall our power rule. If we have X to the exponent M to the exponent N, you can write this as X to the exponent M multiplied by N. OK. So an exponent raised to an exponent, we can simplify it into a single exponent by multiplying those original exponents together. So for our, a term, we have a to the exponent nine to the exponent three, we multiply the exponents together. So we get nine multiplied by one third, we have A to the exponent 9/ and doing the same for our B term B to the exponent multiplied by one third, we get B to the exponent 15 3rd. And now we can simplify our exponent in this expression. So we get 6 a to the exponent 9/3 which gives us three B to the exponent 15/3, which is equal to five. And this is a simplified expression, we can't simplify any further. That's what we were looking to find. OK. And we found that the correct answer is six A to the exponent three B to the exponent five which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

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In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. (25x⁴y⁶)^½