Adding and Subtracting Rational Numbers
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6.4 Adding and Subtracting Rational Numbers Addition and subtraction of rational numbers and rational expressions with equal denominators is relatively simple. Rational Numbers Rational Expressions = = = = Rational numbers or rational expressions can be added or subtracted only when there are common denominators. When the denominators are not common, the rational numbers or rational expressions must be re-written so that the denominators are common. This can be accomplished by finding the lowest common multiple (LCM) of all of the denominators, and re-writing each number or expression with this LCM as the denominator.
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Rational Number Operations
Transcript of Adding and Subtracting Rational Numbers
6.4Adding and Subtracting Rational Numbers
Addition and subtraction of rational numbers and rational expressions with equal denominators is relatively simple.
Rational NumbersRational Expressions
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Rational numbers or rational expressions can be added or subtracted only when there are common denominators.
When the denominators are not common, the rational numbers or rational expressions must be re-written so that the denominators are common. This can be accomplished by finding the lowest common multiple (LCM) of all of the denominators, and re-writing each number or expression with this LCM as the denominator.
The lowest common denominator is the lowest number or expression that is the least common multiple of all the denominators.
An example with rational numbers will show this concept in its simplest form.
Example 1
Simplify .
Solution:
Find the lowest common denominator.The LCM of 3, 4 and 8 is 24.
Make equivalent terms with a denominator of 24.
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Combine the numerators.=
Simplify.=
The lowest common denominator (LCD) is the lowest common multiple of the denominators.
The denominators may first need to be factored to make sure all of the factors are included in the LCD.
The same procedure is true when adding and subtracting rational expressions. All the steps must be followed. There are no short cuts. As you become more familiar with the process it will be easier to mentally find the lowest common denominator and simplify the expressions.
Example 2
Simplify .
Solution:
Find the lowest common denominator.The LCM of 6a, 9a and 4a is 36a.
Make equivalent terms with a denominator of 36a.
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Combine the numerators.=
Simplify.=
The numerators of the terms can include binomials or trinomials. In this case, the distributive property must be applied when multiplying to make equivalent terms with a common denominator.
Example 3
Simplify .
Solution:
Find the lowest common denominator.The LCM of 3y and 4y is 12y.
Make equivalent terms with a denominator of 12y.
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Use the distributive property.=
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Combine the numerators.=
Simplify.=
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Whenever you subtract a polynomial, it is important to remember to distribute the negative to all the terms in the polynomial.
Adding and subtracting polynomial expressions with binomial or polynomial denominators involves another step when simplifying these expressions. The denominators have to be factored first in order to make sure that all the factors are included in the lowest common denominator.
Example 4
Find the lowest common multiple of and .
Solution:
Factor each expression.=
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Each factor must be represented in the LCM, but a factor does not need to be repeated if it is already included.
The lowest common multiple is .
common factor the leftovers
Example 5
Simplify .
Solution:
Find the lowest common denominator.The LCM of and is .
Make equivalent terms with a denominator of .
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Combine numerators.=
Distribute the negative.=
Simplify.=
Example 6
Simplify .
Solution:
Find the lowest common denominator by factoring each term in the denominator.
The LCM is .
common factorleftover Make equivalent terms with a denominator of .
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Example 7
Simplify .
Solution:
Sometimes a common denominator can be found by multiplying the numerator and the denominator of an expression by .
Multiply the numerator and the denominator of the second term by .
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Find the lowest common denominator.
The lowest common denominator is .
Make equivalent fractions with a denominator of .
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Combine the numerators.=
Exercise 6.4
1.Find the least common multiple of the following sets of expressions.
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2.State the LCD and simplify.
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3.State the LCD and simplify.
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