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Simplifying Expressions with a Fraction Bar

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Simplifying Expressions with a Fraction Bar

Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction \(-\frac{1}{3}\) could be the result of dividing \(\frac{-1}{3},\) a negative by a positive, or of dividing \(\frac{1}{-3},\) a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.

Simplifying Expressions with a Fraction Bar

If both the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative.

\(\frac{-1}{-3}=\frac{1}{3}\phantom{\rule{4em}{0ex}}\frac{\text{negative}}{\text{negative}}=\text{positive}\)

Placement of Negative Sign in a Fraction

For any positive numbers \(a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b,\)

\(\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\)

Example

Which of the following fractions are equivalent to \(\frac{7}{-8}?\)

\(\phantom{\rule{0.2em}{0ex}}\frac{-7}{-8},\frac{-7}{8},\frac{7}{8},-\frac{7}{8}\)

Solution

The quotient of a positive and a negative is a negative, so \(\frac{7}{-8}\) is negative. Of the fractions listed, \(\phantom{\rule{0.2em}{0ex}}\frac{-7}{8}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-\frac{7}{8}\) are also negative.

Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses. For example, \(\frac{4+8}{5-3}\) means \(\left(4+8\right)÷\left(5-3\right).\) The order of operations tells us to simplify the numerator and the denominator first—as if there were parentheses—before we divide.

We’ll add fraction bars to our set of grouping symbols from Mathematics 102 to have a more complete set here.

Simplifying Expressions with a Fraction Bar

How to Simplify an Expression with a Fraction Bar

  1. Simplify the numerator.
  2. Simplify the denominator.
  3. Simplify the fraction.

Example

Simplify: \(\frac{4+8}{5-3}.\)

Solution

 \(\frac{4+8}{5-3}\)
Simplify the expression in the numerator.\(\frac{12}{5-3}\)
Simplify the expression in the denominator.\(\frac{12}{2}\)
Simplify the fraction.6

Example

Simplify: \(\frac{4-2\left(3\right)}{{2}^{2}+2}.\)

Solution

 \(\frac{4-2\left(3\right)}{{2}^{2}+2}\)
Use the order of operations. Multiply in the numerator and use the exponent in the denominator.\(\frac{4-6}{4+2}\)
Simplify the numerator and the denominator.\(\frac{-2}{6}\)
Simplify the fraction.\(-\frac{1}{3}\)

Example

Simplify: \(\frac{{\left(8-4\right)}^{2}}{{8}^{2}-{4}^{2}}.\)

Solution

 \(\frac{{\left(8-4\right)}^{2}}{{8}^{2}-{4}^{2}}\)
Use the order of operations (parentheses first, then exponents).\(\frac{{\left(4\right)}^{2}}{64-16}\)
Simplify the numerator and denominator.\(\frac{16}{48}\)
Simplify the fraction.\(\frac{1}{3}\)

Example

Simplify: \(\frac{4\left(-3\right)+6\left(-2\right)}{-3\left(2\right)-2}.\)

Solution

 \(\frac{4\left(-3\right)+6\left(-2\right)}{-3\left(2\right)-2}\)
Multiply.\(\frac{-12+\left(-12\right)}{-6-2}\)
Simplify.\(\frac{-24}{-8}\)
Divide.\(3\)

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