Rational Expressions and Expressions

Rational Exponents

 

Rational Exponents are exponents that have a positive or negative exponent and are fractions.

 

n is the index.

 

a is the radicand.

 

The root sign is called the “Radical Sign”

 

 

Consider:

 

i) if “n” is even, then “a” must be greater than or equal to 0. Otherwise the nth root is not a real number (undefined)

 

 

ii) if “n” is an odd number then  “a” can be a negative or positive number

 

 

 

 

Examples

 

 

In this case, the square root of 64 is positive or negative 8 because (8*8=64) and (-8 * -8 =  64).

 

 

In this case, the answer is only (-2) because positive 2 multiplied by itself 3 times would be positive 8 not (-8).

 

 

*Note: when you have a fraction as the exponent, the number on the bottom becomes the “n” value and the base is the “a” value

 

Advanced Example

 

Express with positive exponents only.

 

 

Click Here to Show Answer

 

 

 

 

Rational Expressions

 

When you have Rational Expressions you must state the restrictions. The restrictions are the numbers that “x” or whatever value there is cannot be to insure that the equation is possible and doesn’t become undefined/not a real number.

 

The restrictions are always set before you cancel out any terms. Once the expression is simplified (without cancelling anything out) then the restrictions are set. After that, continue to simplify the equation.

 

Example 1:

 

 

Restrictions:

 

 

Steps:

 

1. Divide out a “4x” from the top of the equation, which leaves you with (x-2y) on the top

 

2. Since this is multiplication on the top and bottom, the “4x” on the top and bottom cancel each other out resulting in the final answer.

 

3. The restrictions are that “x” and “y” cannot equal 0 because if either one did then the bottom would equal 0. When 0 divides any number it is not real/undefined. Therefore that is the restriction.

 

 

Example 2:

 

 

Restrictions:

 

 

Steps:

 

1. Factor the bottom half of the fraction – since it’s a perfect square then it is factored as shown.

 

2. The top half is not factorable therefore that is the simplest form.

 

3. Therefore the restrictions are that x cannot equal positive or negative 1 because that would cause the bottom to equal 0.

 

Advanced Example

 

 

 

Click Here to Show Answer

 

Restrictions: