Perform the indicated operations and/or simplify each expression....

Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. (∛mn • ∛m²) / ∛n²

Accepted Answer

Hello, today we're going to be simplifying the given expression. So what we're given is the cube root of M squared times N times the cube root of M all over the cube root of end to the fifth. Now, in order to simplify this, we want to go ahead and rationalize the denominator. And in order to rationalize the denominator, we're going to go ahead and multiply the numerator and denominator by the cube root of N. Now, the reason why we multiply the numerator and denominator by the cube root of N in the denominator. If we multiply the cube root of N to the fifth times the cube root of N, the product of two cube roots can combine together to give us a single cube. So we have the cube root of N to the fifth times end, which is going to give us end to the sixth. Now we can go ahead and rewrite into the sixth to give us the cube root of N cubed times N cubed. And the cube root allows you to take out any number that's multiplied by itself three times which in this case is going to be N cubed and N cubed. So we can go ahead and simplify this to just give us end times end, which is going to give us and squared, that's going to be the simplified form of the denominator without the cube root. So now what we want to go ahead and do is simplify the numerator. Now, in the numerator, we have a product of three cube root cube roots, which can be multiplied together and combined into a single cube root. So in the numerator, we have the cube root of M square times N times the cube root of M times the cube root of N. If we combine this into a single cube root, we get the cube root of M squared times N times M times N and combining like terms together M squared times M is going to give us M cubed. So we have the Q bert of M cubed and end times end is going to give us and squared. Now, just like before the cube root allows us to take out any number that's multiplied by itself three times which is going to be M cubed in this case. So we can take out M cubed as a single M and that's going to give us M times the cube root of N squared, which is going to be the simplified version of our numerator. Now, if we put this back into the original expression, what we have is M times the cube group of N squared all over and squared. And that's going to be the rationalized form of the given expression. And with that being said, the answer to this problem is going to be, be. 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.

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. (∛mn • ∛m²) / ∛n²

Key Concepts

Properties of Radicals

Simplifying Expressions

Exponents and Roots

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