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Example Question #3 : Rational Expressions
Which of the following is equivalent to ? Assume that denominators are always nonzero.
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, .
The answer is .