How to Simplify Algebraic Expressions Involving Fractions?
Recap
Simplify:
![\[\frac{12}{64}\]](data:image/svg+xml;nitro-empty-id=MjUwOjEyNzE=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTggMzYiIHdpZHRoPSIxOCIgaGVpZ2h0PSIzNiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
Solution
Factorize both numerator and denominator as follows:
![\[\frac{12}{64}=\frac{2 \times 2 \times 3}{2 \times 2 \times 2 \times 2 \times 2 \times 2}=\frac{4 \times 3}{4 \times 16}=\frac{3}{16}\]](data:image/svg+xml;nitro-empty-id=MjUwOjI5MzQ=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMzI4IDM2IiB3aWR0aD0iMzI4IiBoZWlnaHQ9IjM2IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
How to simplify algebraic fractions
The same concept is applicable when solving algebraic fractions. Let us demonstrate that using an example.
Simplify
![\[\frac{2 a+4}{4 a} \text { and } \frac{9+12 b}{6}\]](data:image/svg+xml;nitro-empty-id=MjUwOjQxNjg=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTUwIDM2IiB3aWR0aD0iMTUwIiBoZWlnaHQ9IjM2IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Solution
To simplify the above algebraic fractions, factorize both numerator and denominator by finding a common term. By inspection, we have:
![\[\frac{2 a+4}{4 a}=\frac{2(a+2)}{4(a)}=\frac{a+2}{2 a}\]](data:image/svg+xml;nitro-empty-id=MjUwOjU0MDk=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjA2IDQzIiB3aWR0aD0iMjA2IiBoZWlnaHQ9IjQzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[\frac{9+12 b}{6+3 a}=\frac{3(3+4 b)}{3(2+a)}=\frac{3+4 b}{2+a}\]](data:image/svg+xml;nitro-empty-id=MjUwOjYzMDU=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjI5IDQzIiB3aWR0aD0iMjI5IiBoZWlnaHQ9IjQzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Algebraic fractions are in the simplest form if there is no common factor in the numerator and denominator and no common factor as in the two cases above. The two examples that we have seen above are direct. However, you can get complex problems, but you should not panic. Just follow the same steps and principles to get the solution.
Example 1:
Simplify
![\[\frac{(a+3)(b-1)}{(a-1)(b-1)}\]](data:image/svg+xml;nitro-empty-id=MjUxOjEyNzY=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTA2IDQzIiB3aWR0aD0iMTA2IiBoZWlnaHQ9IjQzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Solution
By inspection, the term (b-1) is common to both numerator and denominator and hence, cancels out. Just simplify the fraction by canceling the common factor as shown:
![\[\frac{(a+3)(b-1)}{(a-1)(b-1)}=\frac{(a+3)}{(a-1)}\]](data:image/svg+xml;nitro-empty-id=MjUxOjI1NDE=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTg4IDQzIiB3aWR0aD0iMTg4IiBoZWlnaHQ9IjQzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Example 1:
Simplify
![\[\frac{2(a+3)(b-1)}{(2 a+6)}\]](data:image/svg+xml;nitro-empty-id=MjUxOjMyOTE=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTE1IDQzIiB3aWR0aD0iMTE1IiBoZWlnaHQ9IjQzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Solution
There is no common factor in the numerator but we have 2 as a common factor in the denominator. So, we factor out 2 to simplify the fraction.
![\[\frac{2(a+3)(b-1)}{(2 a+6)}=\frac{2(a+3)(b-1)}{2(a+3)}\]](data:image/svg+xml;nitro-empty-id=MjUxOjQ1OTc=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjU3IDQzIiB3aWR0aD0iMjU3IiBoZWlnaHQ9IjQzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Notice how we have the factor 2(a+3) as a common factor for both numerator and denominator. We can cancel it out both at the top and bottom to simplify our fraction.
![\[\frac{2(a+3)(b-1)}{(2 a+6)}=\frac{2(a+3)(b-1)}{2(a+3)}=b-1\]](data:image/svg+xml;nitro-empty-id=MjUxOjU3MTA=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMzE5IDQzIiB3aWR0aD0iMzE5IiBoZWlnaHQ9IjQzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Sometimes you need to group like terms to solve an algebraic expression. We will use this example to demonstrate it.
Simplify
![\[\frac{(a+3 b)-(b-a)}{(2 a+6 b)+(a-2 b)}\]](data:image/svg+xml;nitro-empty-id=MjUxOjY3MjY=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTU0IDQzIiB3aWR0aD0iMTU0IiBoZWlnaHQ9IjQzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Solution
We start the simplification process by first breaking the bracket and collecting like terms together.
![\[\frac{(a+3 b)-(b-a)}{(2 a+6 b)+(a-2 b)}=\frac{a+3 b-b+a}{2 a+6 b+a-3 b}=\frac{a+a+3 b-b}{2 a+a+6 b-3 b}=\frac{2 a+2 b}{3 a+3 b}\]](data:image/svg+xml;nitro-empty-id=MjUxOjgzNTU=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNTQ0IDQ0IiB3aWR0aD0iNTQ0IiBoZWlnaHQ9IjQ0IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Our fraction now looks small, but it is not in its simplest form. We continue to factorizing terms at the top and those at the bottom. By inspection, 2 is common in the numerator, and 3 is common in the denominator. Using this information, we can further simplify the fraction.
![\[\frac{2 a+2 b}{3 a+3 b}=\frac{2(a+b)}{3(a+b)}=\frac{2}{3}\]](data:image/svg+xml;nitro-empty-id=MjUxOjk1NjQ=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTgyIDQzIiB3aWR0aD0iMTgyIiBoZWlnaHQ9IjQzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Some algebraic fractions may seem to be already in the simplest form, but a closer look will tell you that they are not. This trick should not beat you. Inspect the fraction in the example below and conclude whether it is in its simplest form or not.
Examples:
![\[\frac{2(a-2 b)}{3(2 b-a)}\]](data:image/svg+xml;nitro-empty-id=MjUxOjEwODAw-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNzEgNDMiIHdpZHRoPSI3MSIgaGVpZ2h0PSI0MyIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
Solution
What was your response? You are right is you said that the fraction is not in its simplest form. The terms in the braces look unidentical, but we can rearrange them by pulling the negative sign in the denominator out of the bracket. Here is how to go about it:
![\[\frac{2(a+2 b)}{3(2 b-a)}=\frac{2(a-2 b)}{-3(a-2 b)}\]](data:image/svg+xml;nitro-empty-id=MjUxOjEyMjE2-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTgyIDQzIiB3aWR0aD0iMTgyIiBoZWlnaHQ9IjQzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Now, look at how nice the fraction has become. The term (a-2b) is now common both in the denominator and numerator. We further simplify the fraction by canceling these terms.
![\[\frac{2(a+2 b)}{3(2 b-a)}=\frac{2(a-2 b)}{-3(a-2 b)}=-\frac{2}{3}\]](data:image/svg+xml;nitro-empty-id=MjUxOjEzMzgw-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjMyIDQzIiB3aWR0aD0iMjMyIiBoZWlnaHQ9IjQzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
That is all for this lesson. You can now simplify any algebraic fractions no matter how complex it seems by just applying these principles. Also, note that you can carry out arithmetic operations on algebraic expressions the same way you do for other numbers and fractions. See you in our next tutorial on how to simplify algebraic expressions involving exponents.
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