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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³√y² 
⁶√y

Accepted Answer

Hello, today we're gonna be simplifying the given expression. So what we are given is the Q root of the quantity three M squared. And that is over the 6th route of the quantity three M. So in order to simplify this, let's go ahead and rewrite the numerator and denominator with rational exponents. Or in other words, we're gonna change the square roots into exponents and we have a property that allows us to 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 inside the square root is going to become the numerator of the new rational exponent. And the outermost number in front of the square root is going to become the denominator of the new rational exponent. So we can apply this property to both the numerator and denominator in the numerator. The innermost exponent of the quantity three M is going to be two. So we're going to allow N to equal to two. Furthermore, the number in front of the square root symbol is three. So M is going to equal to three. So we can rewrite the numerator as the quantity three M raced to the power of 2/3. And now let's go ahead and apply this property to the denominator. So inside of the square root of the denominator, we have the quantity three M and this is going to be raised to the power of one. So we're going to allow N to equal to one furthermore, the outermost number in front of the square root is six. So M is going to equal to six. And if we apply the property to the denominator, we can rewrite the denominator as the quantity three M raised to the power of 1/6. So now what we have is an exponential value over another exponential value. And we can simplify this by using exponential division. And that states that if you have a race to the power of M over a race to the power of N, then you can go ahead and simplify this by taking the difference of the exponents. Or in other words, we can rewrite this quantity as a race to the power of M minus N. So if we apply this property to the current expression, we can go ahead and simplify this to be the quantity three M raised to the power of 2/3 minus 1/6. And now we just need to simplify the given expo exponent. So in the exponent, we have 2/3 minus 1/6. In order to subtract the two fractions, we need a common denominator. And in this case here, the lowest common denominator between three and six is going to be six. So we only need to change the left most fraction. And in order to change 2/3 to have a denominator of six, we can multiply this fraction by 2/2. So 2/3, multiplied by 2/2 is going to give us 4/6. And what we are left with is 4/6 minus 1/6, which is going to simplify to give us 3/6 and three and six are divisible by three. So if we divide the numerator and denominator by three, we can simplify this to give us one half. So the current, the current expression three M race to the power of 2/3 minus 1/6 is going to simplify to give us the quantity three M race to the power of one half. And we can rewrite this back in radical form by applying the radical property one more time. And so in this case, here N is going to equal to one and M is going to equal to two. And if we put this back into the radical property or the radical notation, we can rewrite this exponential value as the square root of three M. 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.

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. __ ³√y² ⁶√y

Key Concepts

Rational Exponents

Radical Notation

Properties of Exponents

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