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Tips for simplifying Algebraic Fractions

By Kathleen Cantor, 03 Apr 2021

An algebraic fraction is any fraction that uses a variable in the numerator or denominator. For example, the variable x in the fraction x/3 makes it an algebraic fraction. In algebraic fractions, you cannot divide by zero. As such, the denominator has certain restrictions. Take a look at the illustrations below.

  • In the fraction 5/x, x cannot be equal to zero (x≠0)
  • In the fraction 2/(x-3), x cannot be equal to three (x≠3)
  • In the fraction 3/(a-b), a-b cannot be equal to 0 (a-b≠0); therefore, a cannot equal b (a≠b)
  • In the fraction 4/(a²b), neither a nor b can equal 0 (a≠0, b≠0)

Simplifying an algebraic fraction means writing the fraction in its most compact and efficient form without necessarily interfering with the original fraction's value. Simplifying makes it easier for you to understand and solve the fraction. Here are some tips.

How to Simplify Algebraic Fractions

  1. Identify whether you will need to multiply, add, subtract, or divide.
  2. To get the same denominator, multiply the denominators.
  3. Multiply the first numerical fraction by the second denominator to get the first numerator.
  4. Multiply the second numerical fraction by the first denominator to get the second numerator.
  5. Now add, subtract, multiply or divide the numerators to get one answer, then divide by the denominator you got in step 2.
  6. Simplify the fraction if divisible.

Algebraic Notation

Now that you can simplify the basic equation, it will be easier to identify the equation's components. The way you write an algebraic equation is known as algebraic notation.

a/b + c/d is an algebraic notation. The algebraic notation has several components, and they include operators, parentheses, coefficients, variables, and exponents. If you are given an equation like X + PX* V2 – (W/X), X is the variable, + is the operators, P is the coefficient, V2 is the exponent, and (W/X) is the parentheses.

X, which we now know as the variable, represents the unknown number. It helps find the missing number in an equation, which is always the algebra goal. Any letter can represent a variable, and the variable is used together with coefficients, but it does not mean the answer will be the same for both variables.

Coefficients group the same variables into one to simplify work, and operators determine how we will solve the problem. If there are several operators in one problem, we use PEMDAS. Parentheses are the same as brackets. So, if you come across a problem with parentheses, solve it first, and if two parentheses are close to each other, for example (v) (w), you will multiply the two. On the other hand, exponents are numbers multiplied by themselves, V2 in this case, is the same as V*V.

Example 1

Solve the following algebraic fraction: a/b +c/d*(z/p - v/w)

We have addition, subtraction, multiplication, and brackets in the above equation. According to PEMDAS, we must start by solving the equations in the parentheses.

  • z/p – v/w = zw /pw – vp /pw

After parentheses, we will add the remaining equation then multiply the answer we got from the parentheses. In both cases, we apply the law of distribution.

  • a/b + c/d = ad/ bd + bc/ bd
  • (ad/bd + bc/bd)*(zw/pw – vp/pw) = adbc/bd x (zw/pw- vp/pw)

Remember, you are simplifying, not solving.

Example 2

Vx – w / Qx -w

You'll find common factors in the numerator and denominator. Therefore we will group the common factors.

  •   V (x – w)/Q(x-w)

You can cancel out similar factors. This results in the final answer being V/Q.

Example 3

Simplify the equation x2/Vy*PY/QX

You will divide (simplify) x2 by QX resulting in X. You will also divide Y by Vy to get V. So, the final answer will be:

  • X/V*P/ Q
  • XP/VQ

The factor not only has to be common but can also be the highest factor. Therefore, if X is the highest common factor among all the values, you use X to divide each number.

Example 4

Given a rectangle has four sides and the measurements include w+x/v and w-p/x, write an expression for its area.

Since we are finding the area, we will directly multiply the given figures. If it were an addition, we would have added all four sides.

  • W + X/V*W - P/X
  • (W + X)/VX and (W - P)/VX
  • W(W - P) +X(W-P)
  • W2 - WP + XW – XP
  • WP + WP
  • (WP)2/VX

Complex Examples Involving Algebraic Fractions

The following two examples require you to use the factorization method and FOIL method, that is, multiplying two binomials. If the denominators are the same, you will use the examples above and if the denominators are not the same and the question is complex, use the two examples stated. In complex problems, use the following steps:

  1. Factorize the denominators
  2. Write down a factor that appears at least once in the denominators, excluding its powers.
  3. In the factors you have written, write down the largest power that appears in the factors.
  4. The power and factor you get will be the least common divisor you will use to multiply the question's denominator and numerator.

Example 1

Simplify the following complex fraction: (P + p/x)/ (P – p/x2)

Our first step is to create a single fraction in both the numerator and denominator. We will add the fractions in the numerator and subtract the denominator's fractions.

  • {P(x/x) + p/x} / {P (x2/x2) – p/x2}
  • {(x/x + p/x)}/ {(x2 + x2) - p/x2}
  • {(x + p)/x} {/(x2-p)/x2}
  • ((x + p)/x}. {(x2/x2 – p)}
  • {(x+p/p)}. {x2/ (x + p) (x – p)}
  •  x / (x – p)

You can choose to go for a more straightforward method: the least common divisor. In this case, the least common divisor is x2. You will multiply x2 in both the numerator and denominator and quickly solve it.

Example 2

Solve the complex algebraic equation {p/x – p/qx}/ {n/x – w/rx}.

We will first create a single fraction in the denominator and numerator, and afterward, solve the equation.

  • (p/x – p/qx)/ (n/x – w/rx) = {p/x. (q/q) -p/qx}/ (n/x(r/r) – w/rx)

You can choose to leave the answer as it is or simplify depending on the given question.

Final Thoughts

The complex questions may seem hard, but once you understand the essential details, such as factorization, it becomes easy to solve. You can also use the least common divisor where applicable or the greatest common denominator. After multiplication, you can expand the problem to simplify the question.

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