# GRE Math : Rational Expressions

## Example Questions

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### Example Question #2401 : Act Math

Which of the following is equivalent to  ? Assume that denominators are always nonzero.

Explanation:

We will need to simplify the expression . We can think of this as a large fraction with a numerator of and a denominator of .

In order to simplify the numerator, we will need to combine the two fractions. When adding or subtracting fractions, we must have a common denominator. has a denominator of , and  has a denominator of . The least common denominator that these two fractions have in common is . Thus, we are going to write equivalent fractions with denominators of .

In order to convert the fraction  to a denominator with , we will need to multiply the top and bottom by .

Similarly, we will multiply the top and bottom of  by .

We can now rewrite as follows:

=

Let's go back to the original fraction . We will now rewrite the numerator:

=

To simplify this further, we can think of as the same as  . When we divide a fraction by another quantity, this is the same as multiplying the fraction by the reciprocal of that quantity. In other words, .

=

Lastly, we will use the property of exponents which states that, in general, .

### Example Question #1 : How To Divide Rational Expressions

Simplify:

Explanation:

Multiply by the reciprocal of .

Factor

Divide by common factors.

### Example Question #2 : Rational Expressions

Solve for :

Explanation:

To tackle this problem, you need to invert and multiply:

Here we see that we have created a quadratic equation. Therefore, we get all terms to one side, set it equal to zero and use the quadratic formula to solve.

where

Plugging these values in we get the following:

### Example Question #3 : Rational Expressions

Express the following as a single rational expression:

Explanation:

To divide one rational expression by another, invert and multiply:

Remember to foil the numerator meaning, multiply the first components of each binomial. Then multiply the outer components of each binomial. After that, multiply the inner components together, and lastly, multiply the components in the last position of the binomials together.

This arrives at the following:

### Example Question #1 : How To Add Rational Expressions With A Common Denominator

Simplify the following rational expression: (9x - 2)/(x2) MINUS (6x - 8)/(x2)

Explanation:

Since both expressions have a common denominator, x2, we can just recopy the denominator and focus on the numerators. We get (9x - 2) - (6x - 8). We must distribute the negative sign over the 6x - 8 expression which gives us 9x - 2 - 6x + 8 ( -2 minus a -8 gives a +6 since a negative and negative make a positive). The numerator is therefore 3x + 6.

### Example Question #11 : Expressions

Simplify the following rational expression:

Explanation:

Since both fractions in the expression have a common denominator of , we can combine like terms into a single numerator over the denominator:

### Example Question #4 : How To Add Rational Expressions With A Common Denominator

Simplify the following expression:

Explanation:

Since both terms in the expression have the common denominator , combine the fractions and simplify the numerators:

### Example Question #3 : How To Add Rational Expressions With A Common Denominator

Simplify the following rational expression:

Explanation:

Since both rational terms in the expression have the common denominator , combine the numerators and simplify like terms:

### Example Question #4 : Rational Expressions

Explanation:

When adding rational expressions with common denominators, you simply need to add the like terms in the numerator.

### Example Question #5 : Rational Expressions

Choose the answer which best simplifies the following expression: