# SAT II Math I : Solving Rational and Fractional Functions

## Example Questions

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### Example Question #1 : Solving Rational And Fractional Functions

Simplify:

Explanation:

Raise a fraction to a negative power by raising its reciprocal to the power of the absolute value of the exponent. Then apply the properties of exponents as follows:

### Example Question #2 : Solving Rational And Fractional Functions

Simplify

Explanation:

To simplify this problem we need to find the least common denominator between the two fractions. To do this we look at 5 and at 8. The least common number between these two is 40.

In order to rewrite each fraction in terms of a denominator of 40 we need to muliple as follows:

we are able to mulitply by 8/8 and 5/5 because those fractions are really just 1 written in a different format.

Now using order of opperations we get the following

Now we have a common denominator and can do our addition to get the simplfied number:

### Example Question #3 : Solving Rational And Fractional Functions

Solve the following equation to find

Explanation:

In order to be able to find , we must first find the least common denominator. In this case, it is

The equation can now be written as:

Solving for , we get:

### Example Question #6 : Least Common Denominator In Fractions

What's the least common denominator of  and ?

Explanation:

When finding the least common denominator, the quickest way is to multiply the numbers out.

In this case  and  share a factor other than  which is . We can divide those numbers by  to get  and  leftover.

Now, they don't share a common factor so basically multiply them out with the shared factor. Answer is .

Another approach is to list out the factors of both number and find the factor that appears in both sets first.

We can see that  appears in both sets before any other number thus, this is our answer.

### Example Question #4 : Solving Rational And Fractional Functions

Simplify the following:

Explanation:

To simplify the sum of the two fractions, we must find the common denominator.

Simplifying the denominator of the first fraction, we get

because the denominator is a difference of two squares, which follows the form

Now, we can rewrite the sum as

It is far easier to see the common denominator now:

### Example Question #5 : Solving Rational And Fractional Functions

Solve:

Explanation:

To solve this equation, we must find the least common denominator to add the fractions.

Keep in mind that the denominator of the second term is the difference of squares, which can be rewritten as

This is the least common denominator.

Now, we multiply both sides of the equationby the LCD on top and bottom (this is essentially 1):

After canceling terms, we get

Now, we solve for x:

### Example Question #5 : Solving Rational And Fractional Functions

Simplify

Explanation:

Find the least common denominator (LCD) and convert each fraction to the LCD and then add.  Simplify as necessary.

The result is an improper fraction because the numerator is larger than the denominator and can be simplified and converted to a mix numeral.

### Example Question #6 : Solving Rational And Fractional Functions

Find the simplified result:

Explanation:

Start by making both fractions into the same denominator. One option is

Then adjust the numerators by multiplying each fraction's numerator by the other fraction's denominator:

Then we simplify by dividing both numerator and denominator by 2:

which gives us the final result.

### Example Question #6 : Solving Rational And Fractional Functions

Simplify:

Explanation:

To divid fractions, you need to multiply the first fraction by the reciprocal of the second.

Now, multiply the numerators and denominators together, then simplify.

### Example Question #7 : Solving Rational And Fractional Functions

Simplify:

Explanation:

To divide fractions, you need to multiply the first fraction by the reciprocal of the second.

Now, multiply the numerators and denominators, then simplify.

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