Can You Have a Negative in the Denominator
Learning Outcomes
- Place negative fractions that are equivalent given that their negative sign is in a unlike location
- Simplify expressions that contain fraction bars using the guild of operations
Where does the negative sign go in a fraction? Unremarkably, the negative sign is placed in forepart of the fraction, but you will sometimes meet a fraction with a negative numerator or denominator. Remember that fractions represent partition. The fraction [latex]-\frac{1}{three}[/latex] could be the upshot of dividing [latex]\frac{-one}{3}[/latex], a negative by a positive, or of dividing [latex]\frac{1}{-iii}[/latex], a positive past a negative. When the numerator and denominator have unlike signs, the caliber is negative.
If both the numerator and denominator are negative, then the fraction itself is positive because nosotros are dividing a negative past a negative.
[latex]\frac{-1}{-iii}=\frac{i}{3}\frac{\text{negative}}{\text{negative}}=\text{positive}[/latex]
Placement of Negative Sign in a Fraction
For any positive numbers [latex]a\text{ and }b[/latex],
[latex]\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}[/latex]
Example
Which of the following fractions are equivalent to [latex]\frac{7}{-8}?[/latex]
[latex]\frac{-7}{-eight},\frac{-7}{8},\frac{7}{8},-\frac{vii}{8}[/latex]
Solution:
The quotient of a positive and a negative is a negative, then [latex]\frac{7}{-8}[/latex] is negative. Of the fractions listed, [latex]\frac{-7}{8}\text{and}-\frac{7}{8}[/latex] are also negative.
Simplifying an Expression With a Fraction Bar
Fraction bars act every bit grouping symbols. The expressions above and beneath the fraction bar should exist treated equally if they were in parentheses. For example, [latex]\frac{4+eight}{5 - three}[/latex] ways [latex]\left(four+8\right)\div \left(5 - three\correct)[/latex]. The order of operations tells united states to simplify the numerator and the denominator beginning—equally if there were parentheses—earlier we divide.
Nosotros'll add fraction bars to our set of grouping symbols from Utilize the Linguistic communication of Algebra to have a more complete gear up hither.
Grouping Symbols
Simplify an expression with a fraction bar
- Simplify the numerator.
- Simplify the denominator.
- Simplify the fraction.
Instance
Simplify: [latex]\frac{4+eight}{5 - 3}[/latex]
the following video provides some other example of how to simplify diverse expressions that contain a fraction bar.
Example
Simplify: [latex]\frac{iv - 2\left(3\right)}{{2}^{2}+two}[/latex]
Case
Simplify: [latex]\frac{{\left(viii - four\correct)}^{two}}{{eight}^{2}-{iv}^{2}}[/latex]
Example
Simplify: [latex]\frac{4\left(-3\correct)+6\left(-ii\right)}{-iii\left(2\right)-2}[/latex]
Sentinel this video to see another instance of how to simplify an expression with a fraction bar that contains several unlike operations.
Source: https://courses.lumenlearning.com/prealgebra/chapter/simplifying-an-expression-with-a-fraction-bar/
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