Rational Exponents

Rational exponents can be viewed in two ways:

Firstly, if you have a surd (the root of a number that cannot be simplified – for example, the square root of 2) in the denominator of your fraction, you can rationalize the denominator (make it into an integer) by multiplying it by itself. But remember what you do to the bottom you must do to the top.

For example: 2/(square root of 2). Multiply both the top and the bottom by the square root of 2. Your answer will now 2 times square root 2 over 2 which can be simplified to simply square root two by cancelling the 2 in the top and in the bottom.

The second way to view rational exponents to look at the roots as fractions (remember the rule that says the exponent inside the root sign is divided by the value of the root. For example anything inside a square root would be divided by 2 or any exponent inside a cube-root would be divided by 3 and so on.

If we again look at our previous example and changed the roots to exponents, we would now have 2 to the power of 1 over 2 to the power of a half. Now our exponent rules are simple: if I have the same base and I divide I must subtract my denominator’s exponent from my numerator’s exponent.

This means 1 – 1/2 = 1/2.

So I have 2 to the power of 1/2 left or square root 2. This is the same answer as we got by looking at it the first way. The second way might be easier especially if there is more than one variable or number you are working with in the fraction. check out the grade 10 Algebra study guide for some more help with exponents and fractions!