11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;
33, distributive property;
26, distributive property;
$\text{\hspace{0.17em}}\frac{4}{9},$ commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;
0, distributive property, inverse property of addition, identity property of addition
So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as
$\text{\hspace{0.17em}}x+5,\frac{4}{3}\pi {r}^{3},$ or
$\text{\hspace{0.17em}}\sqrt{2{m}^{3}{n}^{2}}.\text{\hspace{0.17em}}$ In the expression
$\text{\hspace{0.17em}}x+5,$ 5 is called a
constant because it does not vary and
x is called a
variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An
algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.
We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.
In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.
Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.
Describing algebraic expressions
List the constants and variables for each algebraic expression.
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.