In Exercises 79–112, use rational exponents to simplify each expr...

Question

In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers.
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⁴√a²b ⋅ ³√ab

Accepted Answer

Hello, today we're gonna be simplifying the given expression. So what we are given is the 4th route of the quantity M N squared. And we're multiplying that by the Q root of the quantity M times N. So in order to simplify this, let's go ahead and rewrite the given square roots as an exponential value. And we have a property that helps us do that. If we have the MTH root of a race to the power of N, then we can rewrite this quantity as a race to the power of N over M here. The innermost exponent N is going to become the numerator of the new rational exponent. And the number in front of the square root symbol M is going to become the denominator of the rational exponent. So if we apply this property to the left hand side of the expression, in this case here, the quantity M N squared is raised to the power of one. So N is going to equal to one. And the number in front of the square root symbol is four. So M is equal to four and we can rewrite the left hand side of the expression as the quantity M N squared race to the power of 1/4. Now let's go ahead and apply the property to the right hand side. The quantity M N is raised to the power of one. So N is going to equal to one and the number in front of the square root symbol is three. So M is equal to three and we can rewrite this as the quantity M N rates the power of 1/3. Now, what we want to go ahead and do is distribute the exponents to each of the terms. So on the left hand side, we're going to distribute 1/4 to M and N squared. And on the right hand side, we're going to distribute one third to M and N. And doing this is going to give us M breaks to the power of 1/ times the quantity N squared, raise the power of 1/4 times M race to the power of 1/3 times N race to the power of 1/3. We need to go ahead and simplify this term right here. And we can simplify this by using our exponential multiplication that states that if you have a race to the power of N race to the power of M, then you can take the product of the exponents and rewrite this quantity as a race to the power of N times M. So here we can rewrite this as N race to the power of two times, 1/4, which is going to give us 2/4 and 2/4 is going to simplify to one half. So we have N race the power of one half. So just to rewrite the entire quantity together, we have M race to the power of 1/4 times. N race to the power of a half times. M race to the power of one third times N race to the power of one third. Now to further simplify this expression, we can go ahead and multiply the M terms together and the end terms together. And we can simplify this by using our exponential edition. And that states that if you have the quantities, a race to the power of M times a race to the power of N, as long as these exponential values have the same base A, then we can go ahead and rewrite this as a race to the power of M plus N. So if we multiply the two M terms together, what we are left with is M raced to the power of 1/ plus 1/3. And if we apply this property to the end terms, we can combine the two end terms together to give us N raised to the power of one half plus 1/3. Now we just need to simplify the, given the given exponents. So in the M term we have 1/4 plus 1/ and 4 and 3 have a common multiple of 12. So we have a lowest common denominator of 12. So if we multiply 1/4 by 3/3 and one third by 1/4, that is going to give us the denominators of 12. And what we are left with is 3/12 plus 1/12. I'm sorry, 4/12 and 3/12 plus 4/ is going to give us 7/12. So the M term is going to simplify to give us M race to the power of 7/12. And on the right hand side for the end terms, we have one half plus one third in the exponent and two in three share a lowest common denominator of six. So if you multiply one half by 3/3 and multiply one third by 2/2, that is going to give us the lowest common denominator of six and a half multiplied by 3/3 is going to give us 3/6 and one third multiply by 2/2 is going to give us 2/6 and 3/6 plus 2/6 is going to give us 5/6. So the end term is going to simplify to end race to the power of 5/6. And this is going to be the simplified form of the given expression and with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers. _ ⁴√a²b ⋅ ³√ab