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Simplify Expressions

Simplifying expressions refers to the concept of simplifying mathematical expressions that have exponents. Learn more from our resources below.

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Simplifying Algebraic Expressions

Corey has a bowl of fruit that consists of 5 apples, 4 oranges, and 3 limes. Katelyn went to the farmer's market and picked up 2 apples, 5 limes, and an orange. How many apples, oranges, and limes do Corey and Katelyn have combined?

Combining like terms is much like grouping together different fruits, like apples and oranges.

Combining Like Terms

Sometimes variables and numbers can be repeated within an expression. If the same variable is in an expression more than once, they can be combined by addition or subtraction. This process is called combining like terms.

 

Simplify each expression

Simplify \begin{align*}5x-12-3x+4\end{align*}5x123x+4.

Reorganize the expression to group together the \begin{align*}x\end{align*}x’s and the numbers. You can either place the like terms next to each together or place parenthesis around the like terms.

\begin{align*}& 5x-12-3x+4\\ & 5x-3x-12+4 \ or \ (5x-3x)+(-12+4)\\ & 2x-8\end{align*}5x123x+45x3x12+4 or (5x3x)+(12+4)2x8

Notice that the Greatest Common Factor (GCF) for \begin{align*}2x\end{align*}2x and 8 is 2. Therefore, you can use the Distributive Property to pull out the GCF to get \begin{align*}2(x-4)\end{align*}2(x4).

Simplify \begin{align*}6a-5b+2a-10b+7\end{align*}6a5b+2a10b+7

Here there are two different variables, \begin{align*}a\end{align*}a and \begin{align*}b\end{align*}b. Even though they are both variables, they are different variables and cannot be combined. Group together the like terms.

\begin{align*}& 6a-5b+2a-10b+7\\ & (6a+2a)+(-5b-10b)+7\\ & (8a-15b+7)\end{align*}6a5b+2a10b+7(6a+2a)+(5b10b)+7(8a15b+7)

There is only one number term, called the constant, so we leave it at the end. Also, in general, list the variables in alphabetical order.

Simplify \begin{align*}w^2+9-4w^2+3w^4-7w-11\end{align*}w2+94w2+3w47w11.

Here we have one variable, but there are different powers (exponents). Like terms must have the same exponent in order to combine them.

\begin{align*}& w^2+9-4w^2+3w^4-7w-11\\ & 3w^4+(w^2-4w^2)-7w+(9-11)\\ & 3w^4-3w^2-7w-2\end{align*}w2+94w2+3w47w113w4+(w24w2)7w+(911)3w43w27w2

When writing an expression with different powers, list the powers from greatest to least, like above.

Examples

Example 1

Earlier, you were asked how many apples, oranges, and limes Corey and Katelyn have, combined.

Let's rewrite Corey's bowl of fruit as \begin{align*}5a+4o+3l\end{align*}5a+4o+3l, where a represents apples, o represents oranges, and l represents limes. Then Katelyn's bowl of fruit can be represented as \begin{align*}2a+5l+o\end{align*}2a+5l+o. Combining like terms, we have:

\begin{align*}(5a+4o+3l)+(2a+5l+o)\\ (5a+2a)+(4o+o)+(3l+5l)\\ 7a+5o+8l\end{align*}(5a+4o+3l)+(2a+5l+o)(5a+2a)+(4o+o)+(3l+5l)7a+5o+8l

Together they have 7 apples, 5 oranges, and 8 limes.

Simplify the expressions below

Example 2

\begin{align*}6s-7t+12t-10s\end{align*}6s7t+12t10s

Combine the \begin{align*}s\end{align*}s’s and the \begin{align*}t\end{align*}t’s.

\begin{align*}& 6s-7t+12t-10s\\ & (6s-10s)+(-7t+12t)\\ & -4s+5t\end{align*}6s7t+12t10s(6s10s)+(7t+12t)4s+5t

Example 3

\begin{align*}7y^2-9x^2+y^2-14x+3x^2-4\end{align*}7y29x2+y214x+3x24

Group together the like terms.

\begin{align*}& 7y^2-9x^2+y^2-14x+3x^2-4\\ & (-9x^2+3x^2)+(7y^2+y^2)-14x-4\\ & -6x^2+8y^2-14x-4\end{align*}7y29x2+y214x+3x24(9x2+3x2)+(7y2+y2)14x46x2+8y214x4

Notice in example 2 that we did not write \begin{align*}(6s-10s)-(7t+12t)\end{align*}(6s10s)(7t+12t) in the second step. This would lead us to an incorrect answer. Whenever grouping together like terms, if one is negative (or being subtracted), always change the operator to addition and make the subtracted number negative.

In example 4, we can also take out the Greatest Common Factor of -2 from each term using the Distributive Property. This would reduce to \begin{align*}-2(3x^2-4y^2+7x+2)\end{align*}2(3x24y2+7x+2). In this case, we take out a -2 so that the first term is positive.

Review

Simplify the following expressions as much as possible. If the expression cannot be simplified, write “cannot be simplified.”

  1. \begin{align*}5b-15b+8d+7d\end{align*}5b15b+8d+7d
  2. \begin{align*}6-11c+5c-18\end{align*}611c+5c18
  3. \begin{align*}3g^2-7g^2+9+12\end{align*}3g27g2+9+12
  4. \begin{align*}8u^2+5u-3u^2-9u+14\end{align*}8u2+5u3u29u+14
  5. \begin{align*}2a-5f\end{align*}
  6. \begin{align*}7p-p^2+9p+q^2-16-5q^2+6\end{align*}
  7. \begin{align*}20x-6-13x+19\end{align*}
  8. \begin{align*}8n-2-5n^2+9n+14\end{align*}

Find the GCF of the following expressions and use the Distributive Property to simplify each one.

  1. \begin{align*}6a-18\end{align*}
  2. \begin{align*}9x^2-15\end{align*}
  3. \begin{align*}14d+7\end{align*}
  4. \begin{align*}3x-24y+21\end{align*}

Challenge We can also use the Distributive Property and GCF to pull out common variables from an expression. Find the GCF and use the Distributive Property to simplify the following expressions.

  1. \begin{align*}2b^2-5b\end{align*}
  2. \begin{align*}m^3-6m^2+11m\end{align*}
  3. \begin{align*}4y^4-12y^3-8y^2\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 1.5 

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Vocabulary

constant

A constant is a value that does not change. In Algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable such as x, y or a.

Greatest Common Factor

The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.

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