SAT Math : How to divide rational expressions

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Example Questions

Example Question #2 : How To Divide Rational Expressions

Which of the following is equivalent to \dpi{100} \frac{(\frac{1}{t}-\frac{1}{x})}{x-t} ? Assume that denominators are always nonzero.

Possible Answers:

(xt)^{-1}

x^{2}-t^{2}

t-x

x-t

\frac{x}{t}

Correct answer:

(xt)^{-1}

Explanation:

We will need to simplify the expression \frac{(\frac{1}{t}-\frac{1}{x})}{x-t}. We can think of this as a large fraction with a numerator of \frac{1}{t}-\frac{1}{x} and a denominator of \dpi{100} x-t.

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. \frac{1}{t} has a denominator of \dpi{100} t, and \dpi{100} -\frac{1}{x} has a denominator of \dpi{100} x. The least common denominator that these two fractions have in common is \dpi{100} xt. Thus, we are going to write equivalent fractions with denominators of \dpi{100} xt.

In order to convert the fraction \dpi{100} \frac{1}{t} to a denominator with \dpi{100} xt, we will need to multiply the top and bottom by \dpi{100} x.

\frac{1}{t}=\frac{1\cdot x}{t\cdot x}=\frac{x}{xt}

Similarly, we will multiply the top and bottom of \dpi{100} -\frac{1}{x} by \dpi{100} t.

\frac{1}{x}=\frac{1\cdot t}{x\cdot t}=\frac{t}{xt}

We can now rewrite \frac{1}{t}-\frac{1}{x} as follows:

\frac{1}{t}-\frac{1}{x} = \frac{x}{xt}-\frac{t}{xt}=\frac{x-t}{xt}

Let's go back to the original fraction \frac{(\frac{1}{t}-\frac{1}{x})}{x-t}. We will now rewrite the numerator:

\frac{(\frac{1}{t}-\frac{1}{x})}{x-t} = \frac{\frac{x-t}{xt}}{x-t}

To simplify this further, we can think of \frac{\frac{x-t}{xt}}{x-t} as the same as \frac{x-t}{xt}\div (x-t) . 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, a\div b=a\cdot \frac{1}{b}.

 

\frac{x-t}{xt}\div (x-t) = \frac{x-t}{xt}\cdot \frac{1}{x-t}=\frac{x-t}{xt(x-t)}= \frac{1}{xt}

Lastly, we will use the property of exponents which states that, in general, \frac{1}{a}=a^{-1}.

\frac{1}{xt}=(xt)^{-1}

The answer is (xt)^{-1}.

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