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Rational Exponents
The laws for rational exponents work exactly the same as the laws for integer exponents. Here is a review of the laws of exponents
Exponent Laws
Product Law Power to a Power Product to a Power Quotient Law Quotient to a Power
All rational exponents can be rewritten as radicals. So, your main question is probably, "How do I convert a rational expoent to a radical?" We are so glad you asked.
Let's look at this problem
To convert a rational exponent to a radical the denominator becomes the index(the number in the crevest) and the numerator of the exponent stays with the number or varable and is in the radicand
So =
You notice how there is no number in the crevest. When there is no number in the crevest. It is understood to be a 2.
Lets try coverting more rational exponents to radicals
Example 1
What number becomes the index of the radical?
What number can multiply 4 times in a row to get 81?
= 3
Example 2
What radical will be equal to this rational exponent?
here we find the square root of the numerator and the square root of the denominator.
=
So much fun!
Example 3
What radical will be equal to this rational exponent?
To find the square root of negative 4 we must ask ourselves what two identical numbers multiply together to give us 4. Hmmmm. Lets see if we can find a pair. and Well, the truth is that there are no real numbers that we can square and get 4. Thus, the answer is "no real solution"
Example 4
The difference between this example and the previous example is the following. In the previous example negative 4 was INSIDE parenthesis. In this example the negative is outside parenthesis, and thus the fractional exaponent is only applied to the 4 not the negative sign.
What radical will be equal to this rational exponent?
= 2
Example 5
What radical will be equal to this rational exponent?
First get the square root of 4 which is 2
Then raise 2 to the 5^{th} power.
= 32
Good job!
Example 6
Here we have a negative exponent. Well using the rules of exponents, a negative exponent causes the fraction to shift. What is in the numerator moves to the denominator and vise versa.
=
We have seen a few half powers by now so we should know that this converts to the square root.
=
Pretty easy huh?
Example 7
Here we have a negative exponent. Well using the rules of exponents, a negative exponent causes the fraction to shift. What is in the numerator moves to the denominator and vise versa.
=
Next, we convert this to a radical.
The cube root of negative 27 is 3
= 9
Example 8
The laws of exponents tells us that we should add the fractions.
=
Rewind to prealgebra. To add fractions of different denominators we need a common denominator In this case the common denominator is 6.
= = =
What radical will be equal to this rational exponent?
=
The cube root of 8 is 2
= 4
Not bad.
Example 9
What does the laws of exponents tells us to do with the exponents?
Subtract. Bring the smaller number to the greater number and subtract.
We need a common denominator. The common denominator is 20.
Simplify.
=
In math, we do not like to leave radicals in the exponents. So, we will talk about how to do that in the next section.
Example 10
Distribute
Simplify.
= =
So much fun.
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