Evaluating and Simplifying Expressions | Geometry | LevelKite

You will compute expressions all through geometry, whenever you find a length, an area or a volume. A numerical expression combines numbers with operations, such as addition and multiplication, and has no equals sign. To evaluate an expression means to find the one number it names.

Take $2 + 3 \cdot 4$ . Adding first gives $20$ , but multiplying first gives $14$ . The meaning settles it: $3 \cdot 4$ is three groups of $4$ , so the expression says $2 + 4 + 4 + 4$ , which is $14$ . In the same way, a positive whole-number exponent tells how many times to use its base as a factor, so $3^2$ means $3 \cdot 3$ . Exponents are built from multiplication, and whole-number multiplication is built from addition. That is why exponents come before multiplication, and multiplication comes before addition.

The order of operations keeps this pattern for all numbers. It is a convention, meaning an agreement, not a theorem, which is a statement you prove. Because everyone follows it, an expression has the same value for every reader.

The order of operations: work from the top row down.

Rank	Operations	How to work
1	Grouping symbols: parentheses, brackets, fraction bars, square root signs	Finish each group first, innermost first. Inside a group, use this same order.
2	Exponents and roots	An exponent applies only to its base: the number, letter or group in parentheses just before it.
3	Multiplication and division	Same rank: work left to right. A divisor (the number you divide by) must be nonzero; division by zero is undefined.
4	Addition and subtraction	Same rank: work left to right.

Worked example

Evaluating a numerical expression

Evaluate $30 - 12 \div 2 \cdot 3 + (1 + 3)^2$ .

Grouping symbols come first in the order of operations, so add inside the parentheses.
$30 - 12 \div 2 \cdot 3 + 4^2$
Exponents come next. By the definition of an exponent, the base 4 is used as a factor twice: 4 times 4 is 16.
$30 - 12 \div 2 \cdot 3 + 16$
Multiplication and division share a rank, so work left to right. The division comes first here. Its divisor, 2, is nonzero, so the division is defined.
$30 - 6 \cdot 3 + 16$
Do the remaining multiplication, because it comes before addition and subtraction.
$30 - 18 + 16$
Addition and subtraction share a rank, so work left to right: subtract first, then add.
$12 + 16 = 28$

Answer

The value of $30 - 12 \div 2 \cdot 3 + (1 + 3)^2$ is $28$ .

A variable is a letter that stands for a number. An algebraic expression is an expression with at least one variable, such as $x^2 - 2y$ . A number written next to a variable means multiplication, so $2y$ means $2 \cdot y$ .

To evaluate an algebraic expression, you substitute: you replace each variable with its given value. The substitution property of equality makes this valid. It says that if two quantities are equal, you may replace one with the other in any expression without changing the expression's value. Write each value in parentheses so that it stays together as one number, and then follow the order of operations.

After you substitute, check each denominator, the bottom part of a fraction. A fraction bar means division, so a denominator is a divisor and must be nonzero. If a denominator equals $0$ , the expression is undefined for those values, which means it has no value to report.

Worked example

Substituting a negative value

Given $x = -4$ and $y = 2$ , evaluate $\dfrac{x^2 - 2y}{x + y}$ .

By the substitution property of equality, replace x with -4 and y with 2. Parentheses keep each value together as one number.
$\dfrac{(-4)^2 - 2(2)}{(-4) + (2)}$
A fraction bar is a grouping symbol, so finish the numerator (the top) and the denominator (the bottom) separately. In the numerator, the exponent comes first: -4 times -4 is 16, because the product of two negative numbers is positive.
$\dfrac{16 - 2(2)}{(-4) + (2)}$
In the numerator, multiply before you subtract.
$\dfrac{16 - 4}{(-4) + (2)} = \dfrac{12}{(-4) + (2)}$
Add in the denominator to get -2. This is nonzero, so the fraction is defined. If y were 4 instead, the denominator would be 0 and the expression would be undefined.
$\dfrac{12}{-2}$
Divide by the nonzero denominator. A positive number divided by a negative number is negative.
$\dfrac{12}{-2} = -6$

Answer

When $x = -4$ and $y = 2$ , the expression has the value $-6$ .

Key concept

One expression, one value

Once every variable has a value and every operation is defined, which requires every denominator to be nonzero, an expression names exactly one number. Substitution puts the values in, and the order of operations makes that number the same for everyone. The rest of this lesson builds on this idea: measures in geometry come from expressions like these, and rewriting an expression must never change the number it names.

A formula is an equation that shows how to compute one quantity from others. You evaluate a formula the same way you evaluate any algebraic expression, with substitution and the order of operations. The difference is that the values are now measures, so three more habits apply:

Use only values that are given or marked. Never take a measure from how a drawing looks.
Check that each length, area and volume is positive, and that each denominator is nonzero. A coordinate may be negative, because it names a position rather than a measure.
Keep the units. Lengths use units such as $\text{cm}$ , areas use square units such as $\text{cm}^2$ , and volumes use cubic units such as $\text{cm}^3$ .

The symbol $\pi$ is not a variable. It names one fixed number, about $3.14159$ . Keep it as a factor for an exact answer, or replace it with a decimal for an approximate answer.

Worked example

Evaluating a volume formula

Given: a cone has a circular base with radius $r = 3$ cm and a height of $h = 8$ cm, measured perpendicular to the base. Find: its volume from $V = \frac{1}{3}\pi r^2 h$ , as an exact value and rounded to the nearest tenth.

Check the given values. A radius and a height are lengths, so both must be positive, and they are. The only denominator, 3, is nonzero.
$r = 3 > 0, \quad h = 8 > 0$
Substitute the values, using the substitution property of equality.
$V = \frac{1}{3}\pi(3)^2(8)$
The exponent comes before multiplication in the order of operations.
$V = \frac{1}{3}\pi(9)(8)$
Only multiplication remains. The commutative and associative properties of multiplication let you reorder and group the number factors.
$V = \left(\frac{1}{3} \cdot 9 \cdot 8\right)\pi = (3 \cdot 8)\pi = 24\pi$
Keep the exact value, then replace pi with about 3.14159 and round to the nearest tenth. Square centimeters times centimeters give cubic centimeters. The result is positive, as a volume must be.
$V = 24\pi\text{ cm}^3 \approx 75.4\text{ cm}^3$

Answer

The cone's volume is exactly $24\pi$ cubic centimeters, which is about $75.4$ cubic centimeters.

Other formulas, same routine. Values are given, denominators are nonzero, and each result is a positive measure.

Measure	Formula	Given values	Substitute	Evaluate
Perimeter of a rectangle with length $\ell$ and width $w$	$P = 2(\ell + w)$	$\ell = 7$ m, $w = 4$ m	$P = 2(7 + 4)$	$P = 2(11) = 22$ m
Area of a triangle with base $b$ and height $h$ to that base	$A = \frac{1}{2}bh$	$b = 10$ cm, $h = 6$ cm	$A = \frac{1}{2}(10)(6)$	$A = 5(6) = 30\text{ cm}^2$
Area of a circle with radius $r$	$A = \pi r^2$	$r = 5$ m	$A = \pi(5)^2$	$A = 25\pi\text{ m}^2 \approx 78.5\text{ m}^2$
Distance between points $(x_1, y_1)$ and $(x_2, y_2)$	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	$(x_1, y_1) = (-1, 2)$ , $(x_2, y_2) = (2, 6)$	$d = \sqrt{(2 - (-1))^2 + (6 - 2)^2}$	$d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ units

You can also rewrite an expression without changing the number it names. Two expressions are equivalent when they have the same value for all values of the variables where both are defined, so no denominator equals $0$ . To simplify an expression in this lesson means to write an equivalent expression with no parentheses left to multiply out and with all like terms combined.

A term is a number, a variable, or a product of numbers and variables. Addition and subtraction signs outside grouping symbols separate terms, and a subtraction sign stays with the term after it, so $6x - 4x$ has the terms $6x$ and $-4x$ . The number factor in a term is its coefficient. Like terms have the same variables with the same exponents. You can combine them because they count the same kind of thing: $3x + 2x$ is three groups of $x$ plus two groups of $x$ , which makes five groups of $x$ , or $5x$ .

Terms	Like terms?	Reason
$6x$ and $-4x$	Yes	Same variable, same exponent: $x$ to the first power
$2x^2$ and $7x^2$	Yes	Same variable, same exponent: $x$ to the second power
$5$ and $-3$	Yes	Both are numbers with no variable
$2x^2$ and $2x$	No	Same variable, but the exponents are $2$ and $1$
$3x$ and $3y$	No	Different variables

Distributing has a meaning you can picture. Imagine a rectangle with width $a$ and length $b + c$ , where all three are positive. Its area is $a(b + c)$ . Cut it into two rectangles with the same width $a$ and with lengths $b$ and $c$ . Their areas are $ab$ and $ac$ . The Area Addition Postulate, a statement accepted without proof, says that a region's area is the sum of the areas of its non-overlapping parts. So the two results must match.

This rectangle shows the meaning only for positive lengths. The general fact is the distributive property, a basic property of numbers that is also accepted without proof. It holds for all real numbers $a$ , $b$ and $c$ , which means any numbers on the number line, including negatives and zero:

a(b + c) = ab + ac

Read from left to right, this equation removes parentheses, which is called distributing: $2(x + 3) = 2x + 6$ . Read from right to left, it collects a shared factor, which is how like terms combine: $3x + 2x = (3 + 2)x = 5x$ . You add the coefficients and keep the variable part. The commutative property of multiplication, $ab = ba$ , lets the shared factor stand on either side.

Subtraction fits too. By the definition of subtraction, $b - c = b + (-c)$ , so $a(b - c) = a(b + (-c)) = ab - ac$ .

Worked example

Distributing and combining like terms

Given: the expression $2x(x + 3) - 4(x - 5)$ . Find: an equivalent simplified expression. Then check it at $x = 2$ .

By the definition of subtraction, subtracting a number means adding its opposite. So the outside factor is -4, and x - 5 is x + (-5).
$2x(x + 3) + (-4)(x + (-5))$
Use the distributive property from left to right on each product.
$2x \cdot x + 2x \cdot 3 + (-4)(x) + (-4)(-5)$
Multiply each pair, grouping number factors by the commutative and associative properties. By the definition of an exponent, x times x is x squared. Two negative factors give a positive product, and adding -4x is written as subtracting 4x.
$2x^2 + 6x - 4x + 20$
The terms 6x and -4x are like terms. Use the distributive property from right to left to combine them. The term with x squared has a different exponent, so it stays separate, and so does the number 20.
$2x^2 + (6 - 4)x + 20 = 2x^2 + 2x + 20$
Check the original expression at x = 2, using substitution and the order of operations.
$2(2)((2) + 3) - 4((2) - 5) = 4(5) - 4(-3) = 20 + 12 = 32$
Evaluate the simplified expression at x = 2. Both values are 32, so the check finds no mistake.
$2(2)^2 + 2(2) + 20 = 8 + 4 + 20 = 32$

Answer

The simplified form of $2x(x + 3) - 4(x - 5)$ is $2x^2 + 2x + 20$ , and the check at $x = 2$ gives $32$ for both forms.

Here is the central idea: an expression names one number once its variables have values and its operations are defined. To find that number, substitute each value in parentheses, check that denominators are nonzero and measures are positive, and follow the order of operations. To simplify, distribute and combine like terms. The new form looks different, but it names the same number wherever both forms are defined.