# Algebra: How Do You Do Rational Exponents?

**Rational Exponents**

Rational exponents are sometimes referred to as a law of indices. There are many laws of indices but this is just one of them. Rational exponents are defined by the following:

When we talk look at the idea of exponents or powers, we normally think of regular numbers such as 3^{2} and 5^{3} but have we every consider the exponent as being a fraction? We will not examine the concept of rational/fractional exponents

Let us say that we have 4^{2} what would be the answer. It was be 4 × 4 = 16. Consider the question: if you raised 4 to the power of 2 to get 16 what do you think you should raise 16 to get back 4. The answer is based on the concept of reciprocal.

This means that if you raise 4 to the power of 2 to get 16 (4^{2 }= 16), then you need to raise 16 to the reciprocal of the power to get 4 therefore 16^{1/2} = 4 where ½ is the reciprocal of 2, and 16^{1/2} = √16.

You should be able to see that raising a number to 1/n where ‘n’ is the nth root is defined as

a^{1/n} = n√a

Let us prove it:

9^{1/2} × 9^{1/2} = 9^{1/2 + 1/2 }= 9^{1} and by doing √9 × √9 = 3 × 3 = 9 you get the same answer. Therefore, 9^{1/2} = √9.

Let us look at 4^{3/2}. Consider this, is 3/2 is the same as 3 × ½. Yes it is.

Therefore: 4^{3/2}= 4^{3 × ½} = (4^{3})^{1/2} by the same principle, you should note that: (4^{3})^{1/2} = √4^{(3)×1} = √4^{(3) }

Thus, you should appreciate the general term a^{m/n} = ^{n}√a^{m}= (^{n}√a)^{m}, where ‘*n*’ is the “*nth*” root.

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**Example 1:**

Simplify 125^{2/3}. To solve this problem we recall: a^{m/n }= (^{n}√a)^{m}

Therefore: 125^{2/3 }= (3√125)^{2 }

= (5)^{2}

= 25

You should also note that a certain root of number can be negative and you can find the root of certain negative values. In mathematical principle, you will be told that you cannot find the root of a negative number, root is referred to squared root. But when you do further math such as pre-calculus and calculus you will see that you actually can. Anyway while you cannot find the squared root of a negative number as was mentioned you can find a *certain* root of a negative number.

**Example 2**

What is the 3 cube root of – 27? Cube root is written as a number raised to the power of a third.

Thus -27^{1/3} = 3√-27 = – 3.

But how do we know that is true? We can check it. If -3 is the answer then we should be able to multiply -3 by itself three times and the results should be – 27.

Thus -3 × -3 = 9 × -3 = -27.

To see if you grasp the concept you can try these as practice.

- 25
^{3/2} - 256
^{2/4} - 216
^{2/3} - -32
^{1/5} - (36
*x*^{2})^{1/2}