Common Math Verbs
The following math verbs are commonly seen in math exercises, particularly in algebra. Unfortunately, the meaning of each is not often taught, and it is often assumed students know what they mean. Understanding the meaning of each of these math verbs can help you approach a question appropriately.
Simplify
In general, to simplify means to put an expression into a form that makes it easier to work with. This may involve several steps, and what you need to do to simplify an expression depends on what you start with.
Simplify a numerical expression
To simplify a numerical expression means to do all the math possible by following the correct order of operations.
Example:
Simplify.
5 + 2³ − 3(4 − 2)5 plus 2 cubed minus 3 open parentheses 4 minus 2 close parentheses
Simplify the expression: 5 plus 2 cubed minus 3 times open bracket 4 minus 2 close bracket.
Line 1: Subtract inside the brackets, so the expression is 5 plus 2 cubed minus 3 times 2
Line 2: Apply the exponent of 3 on the 2, so the expression is 5 plus 8 minus 3 times 2
Line 3: Multiply negative 3 by 2, so the expression is 5 plus 8 - 6
Line 4: Add the 5 and 8, so the expression is 13 minus 6
Line 5: Subtract 6 from 13, so the simplified expression is 7.
Simplify a fraction
To simplify a fraction means to reduce a fraction into lowest terms. To simplify a fraction, divide the numerator and the denominator by the greatest common factor (GCF). A fraction is considered simplified (in lowest terms) if there are no common factors in the numerator and the denominator.
Example 1:
Simplify the following fraction.
Example 1. Simplify the following fraction: 8 over 12.
Line 1: In the fraction 8 over 12, the greatest common factor of the numerator and denominator is 4.
Line 2: Divide the numerator and denominator by 4 so the numerator of the fraction becomes 8 divided by 4 and the denominator becomes 12 divided by 4.
Line 3: Complete the division, so the simplified fraction is 2 over 3.
Example 2:
Simplify the following fraction.
Example 2. Simplify the following fraction: 6xy over 8x
Line 1: In the fraction 6xy over 8x, the greatest common factor of the numerator and denominator is 2x.
Line 2: Divide the numerator and denominator by 2x so the numerator of the fraction becomes 6xy divided by 2x and the denominator becomes 8x divided by 2x
Line 3: Complete the division, so the simplified fraction is 3y over 4.
Simplify an algebraic expression
To simplify an algebraic expression means to reduce the expression into fewer terms to make it easier to work with. It may involve various methods, such as collecting like terms, applying exponent rules, expanding, or factoring. Simplifying an algebraic expression or equation makes it easier to evaluate or solve.
Example 1:
Simplify by collecting like terms.
Example 1. Simplify the expression: 3x squared minus 5x plus 8 minus 2x squared plus 2x plus 1 by collecting like terms
Line 1: Move the like terms next one another so the expression is 3x squared minus 2x squared minus 5x plus 2x plus 8 plus 1.
Line 2: Combine the like terms so the simplified expression is x squared minus 3x plus 9.
Example 2:
Simplify by expanding.
Example 2. Expand and simplify the expression: 2x times open bracket 3x minus 4 close bracket minus open bracket 4x minus 3 close bracket.
Line 1: Expand the first set of brackets by multiplying each term by 2x so the expression is 6x squared minus 8x minus open bracket 4x minus 3 close bracket.
Line 2: Expand the second set of brackets by multiplying each term by -1 so the expression is 6x squared minus 8x minus 4x plus 3.
Line 3: Combine the like terms so the simplified expression is 6x squared minus 12x plus 3.
Example 3:
Example 3. Using exponent rules, simplify the expression x to the power of 5 times x squared all over x cubed.
Line 1: Use the product property of exponents to simplify the expression to x to the power of 7 over x cubed
Line 2: Use the quotient property of exponents to simplify the expression to x to the power of 4
Example 4:
Evaluate
To evaluate an algebraic expression means to find the values of the expression when the variable is replaced with a given number. Before evaluating an expression, check to see if it can be simplified first. Next, substitute the given number for the variable into the expression and simplify using the order of operations.
Example 1:
Evaluate 2x − 4 when x = 32x minus 4
3 lines
Line 1: Substitute in 3 for the x variable. The expression equals 2 times negative minus 4
Line 2: Multiply. The expression equals 6 minus 4 .
Line 3: Subtract to get the final value. The expression equals 2.
Example 2:
If x = −2 and y = 
, evaluate:If x equals minus 2 and y equals 1 third, evaluate:
Evaluate
To evaluate an algebraic expression means to find the values of the expression when the variable is replaced with a given number. Before evaluating an expression, check to see if it can be simplified first. Next, substitute the given number for the variable into the expression and simplify using the order of operations.
Example 1:
Evaluate 2x − when x = 32x minus 4
3 lines
Line 1: Substitute in 3 for the x variable. The expression equals 2 times negative minus 4
Line 2: Multiply. The expression equals 6 minus 4 .
Line 3: Subtract to get the final value. The expression equals 2.
Example 2:
If x = −2 and y = , evaluate:If x equals minus 2 and y equals 1 third, evaluate:
Solve
To solve an algebraic expression means to find a value or set of values for the variable(s) in a given equation by isolating for the variable. An algebraic equation states that two algebraic expressions are equal. Therefore, to solve an equation is to determine the values of the variable(s) that make the equation a true statement. Any number that makes the equation true is called a solution to the equation. If possible, simplify both sides of the equation before solving it. A solution can always be checked by substituting the value back into the given equation and verifying the statement is true.
Example 1:
Solve by isolating for the variable x. Check your solution.
Line 1: Solve the equation 8x plus 9x minus 5x equals negative 3 plus 15
Line 2: Combine like terms to simplify the equation to 12x = 12
Line 3: Divide both sides of the equation by 12 to isolate for x so the equation is 12x divided by 12 equals 12 divided by 12.
Line 4: Simplify to get the solution x equals 1
Check:
Line 1: Check your solution in the given equation 8x plus 9x minus 5x equals negative 3 plus 15
Line 2: Substitute x equals 1 into the given equation so the equation is 8 times 1 plus 9 times 1 minus 5 times 1 equals negative 3 plus 15
Line 3: Multiply to simplify the equation to 8 plus 9 minus 5 equals negative 3 plus 15
Line 4: Simplify both sides of the equation to 12 equals 12
Line 5: Therefore, the left side of the equation equals the right side of the equation, and we have verified the solution x equals 1 is correct.
Example 2:
Solve by isolating for the variable n.
Line 1: Solve the equation negative 3 times open parenthesis n minus 2 close parenthesis minus 6 equals 21
Line 2: Use the distributive property to expand the parentheses so the equation is negative 3n plus 6 minus 6 equals 21
Line 3: Complete the subtraction to simplify the equation to negative 3n equals 21
Line 4: Divide both sides of the equation by negative 3 to isolate for n so the equation is negative 3n divided by negative 3 equals 21 divided by negative 3.
Line 5: Simply to get the solution n equals negative 7.
Line 6: Remember to check your answer by substituting your solution into the given equation and verifying that the left side of the equation equals the right side of the equation.
Remember to check both answers by substituting each solution into the given equation and verifying that the left side of the equation equals the right side of the equation.
Example 3:
Solve for the values of x in the following quadratic equation.
Line 1: Solve for the values x in the quadratic equation x squared plus x minus 6 equals 0
Line 2: Factor the quadratic equation to open parenthesis x minus 2 close parenthesis times open parenthesis x plus 3 close parenthesis equals 0
ine 3: To solve for the first solution, set the first factor to zero so x minus 2 equals 0
Line 4: Isolate for x to get the first solution x equals 2
Line 5: To solve for the second solution, set the second factor to 0 so x plus 3 equals 0
Line 6: Isolate for x to get the second solution x equals negative 3.
Line 7: Therefore, the solutions are x=2 andx= -3.
Line 8: Remember to check your answer by substituting each solution into the given equation and verifying that the left side of the equation equals the right side of the equation.
Remember to check both answers by substituting each solution into the given equation and verifying that the left side of the equation equals the right side of the equation.
Factor
Factoring out the Greatest Common Factor
To factor, or factoring, means to break an expression down into its factors. To factor an expression, first find the greatest common factor (GCF) of all the terms and then rewrite the expression as a product of the GCF. Factoring is the opposite of expanding. An expression is in factored form if the entire expression is written as a product.
Example 1:
Factor the following expressions.
Line 1: Factor the expression 2x plus 14
Line 2: The greatest common factor of 2x and 14 is 2, so we rewrite each term in the expression as a product of the greatest common factor. The expression is 2 times x plus 2 times 7.
Line 3: Factor out the greatest common factor so the expression is 2 times open parenthesis plus 7 close parenthesis.
Example 2:
Factor the following expressions.
Line 1: Factor the expression 14y cubed plus 8y squared minus 10y
Line 2: The greatest common factor of 14y cubed, 8y squared and negative 10y is 2y. Rewrite each term in the expression as a product of the greatest common factor, so the expression is 2y times 7y squared plus 2y times 4y minus 2y times 5.
Line 3: Factor out the greatest common factor so the expression is 2y times open parenthesis 7y squared plus 4y minus 5 close parenthesis.
Factoring a Quadratic Expression
Often you are asked to factor quadratic expressions in the form ax²+bx+c. If a = 1, a quadratic expression in this form can be factored by finding two numbers that multiply together to get the constant c and that sum to the numerical coefficient b.
Example:
Factor the following quadratic expressions.
Factor the quadratic expression x squared plus 3x minus 4.
Step 1: Look for factors of the c value of negative 4 that sum to the b value of 3.
Step 2: Determine that the product of negative 1 and 4 is negative 4 and the sum of negative 1 and 4 is 3. Therefore, the numbers we need are negative 1 and positive 4.
Step 3: Rewrite the expression as a binomial product. Note that because a equals one in the given quadratic expression, the first term in each set of parentheses is x. After each x, fill in the numbers determined from Step 2, so that the expression is open parenthesis x minus 1 close parenthesis times open parenthesis x plus 4 close parenthesis.
Factoring a Quadratic Expression using Grouping
To factor quadratic expressions in the in the form ax² + bx + c when a ≠ 1, a method called grouping can be used.
Example 1:
Fully factor the expression.
Isolate
To isolate means to rearrange an algebraic equation to find a given variable. To isolate for a given variable, perform a series of operations to both sides of the equation until the variable is alone on one side of the equals sign.
Example 1:
Line 1: Isolate for the variable m in the equation E equals m times c squared.
Line 2: Divide both sides of the equation by c squared so the equation is E divided by c squared equals m times c squared divided by c squared.
Line 3: Simplify the expression to E divided by c squared equals m. Therefore, m equals E divided by c squared.
Example 2:
Attribution
Unless otherwise stated, the material in this guide is from the Learning Portal created by College Libraries Ontario. Content has been adapted for the NWP Learning Commons in June 2021. This work is licensed under a Creative Commons BY NC SA 4.0 International License.
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