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.
_
⁴√x
⁵√x

Accepted Answer

Hello, today we're gonna be simplifying the given expression. So what we are given is the cube root of K divided by the fourth root of K. And so in order to simplify this expression, let's go ahead and rewrite the square roots as exponents. And we have a property that helps us do that. The property states that if we have the M 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 becomes the numerator for the new rational exponent. And the outermost number in front of the square root symbol, M becomes the denominator of the new rational exponent. So let's go ahead and apply this property to both the numerator and denominator in the numerator K raise is raised to the power of one. So N is going to equal to one. Furthermore, the number in front of the square root symbol is three. So M is going to equal to three. So if we apply this property to the numerator, we can rewrite the numerator as K raced to the power of 1/3 Now let's go ahead and apply to the denominator in the denominator K is still 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 going to equal to four. So we can rewrite the given denominator as K races to the power of 1/4. Now, what we are left with is an exponential value divided by another exponential value. And we can use our division property for exponential values that property states that if you have a race to the power of M divided by a race to the power of N, then you can simplify this quantity by taking the difference of the exponents. Or in other words, you can rewrite this as a race to the power of M minus N. So if we apply this property to the current, to the current expression, we can rewrite this as K raced to the power of 1/3 minus 1/4. And now we just need to go ahead and simplify the given expressions or the given X exponents. So 1/3 minus 1/4, in order to subtract these two fractions together, we need to have a similar denominator and the lowest common denominator between three and four is going to be 12. So on the left hand side, we can multiply 1/3 by 4/4 to give us a denominator of 12 and 1/4 can be multiplied by 3/3 to get us a denominator of 12. So one third, multiplied by 4/4 is going to give us 4/12 and 1/4 multiplied by 3/3 is going to give us 3/12 and 4/12 minus 3/12 is going to give us 1/12. So the current expression is going to simplify, decay, raced to the power of 1/12. And we can go ahead and rewrite this back into radical form by applying our property going from right to left one more time. So here N is going to equal to one and M is going to equal to 12. And if we apply the exponential property, we can rewrite this as the 12th root of K race to the power of one which is just going to be K 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 D. 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. _ ⁴√x ⁵√x

Key Concepts

Rational Exponents

Radical Notation

Properties of Exponents

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