How do you evaluate fractional exponents?

Answer 1
#x^(a/b) =rootb(x^a) = (rootb(x))^a#

You have two options: either learn why this is the case or just remember this rule:

fractional exponent #1/b#
So first we're going to look at an expression of the form: #x^(1/b)#. To investigate what this means, we need to go from #x to x^(1/b)# and then deduce something from it.
#x^1 = x^(b/b) = x^(1/b*b)# What does multiplication mean? Repeated addition. So we can instead of multiplying by b, adding the number to itself #b# times. #x^(1/b+1/b+1/b+1/b +...)# (b times)
There is a rule you use when multiplying numbers with the same radical: add the exponents. If we reverse this rule, we get: #x^(1/b)*x^(1/b)*x^(1/b)*x^(1/b)*x^(1/b)...# (b times)
Now, we still know that this number is equal to #x#. So now we have to think a bit. What number, multiplied by itself b times, gives you #x#. It's the bth-root of #x# => #x^(1/b)=rootbx#
For example: #8^(1/3)# If we multiply this by itself 3 times we get: #8^(1/3)*8^(1/3)*8^(1/3) = 8^(3/3) = 8# What number multiplied by itself 3 times, gives you 8. It's of course #root3(8) = 2#
What about #a/b#? To know what #x^(a/b)# means, we can further rely on our previous findings: #x^(a/b) = x^(a*1/b) = x^(1/b+1/b+1/b+1/b...) # (a times) #= x^(1/b)*x^(1/b)*x^(1/b)...# (a times)
Repeated multiplication is equal to exponentiation, so we can write: #= (x^(1/b))^a = (rootbx)^a#
You can also bring the exponent in the root: #= rootb(x^a)#
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Answer 2

To evaluate fractional exponents, you can use the property that a fractional exponent represents a root. For example, to evaluate ( x^{\frac{1}{2}} ), you would find the square root of ( x ). Similarly, to evaluate ( x^{\frac{1}{3}} ), you would find the cube root of ( x ), and so on.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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