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# Evaluating Algebraic Expressions

In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

EXAMPLE 1

Evaluate when

1.
2.
Solution

a. To evaluate, substitute for in the expression, and then simplify.

When , the expression has a value of .

b. To evaluate, substitute for in the expression, and then simplify.

When , the expression has a value of .

Notice that we got different results for parts a) and b) even though we started with the same expression. This is because the values used for were different. When we evaluate an expression, the value varies depending on the value used for the variable.

TRY IT 1.1

Evaluate:

1.
2.
1.  10
2.  19

TRY IT 1.2

Evaluate:

1.
2.
1.  4
2.  12

EXAMPLE 2

Evaluate

1.
2.
Solution

Remember means times , so means times .

a. To evaluate the expression when , we substitute for , and then simplify.

 Multiply. Subtract.

b. To evaluate the expression when , we substitute for , and then simplify.

 Multiply. Subtract.

Notice that in part a) that we wrote and in part b) we wrote . Both the dot and the parentheses tell us to multiply.

TRY IT 2.1

Evaluate:

1.  13
2.  5

TRY IT 2.2

Evaluate:

1.  8
2.  16

EXAMPLE 3

Evaluate when .

Solution

We substitute for , and then simplify the expression.

 Use the definition of exponent. Multiply.

When , the expression has a value of .

TRY IT 3.1

Evaluate:

.

64

TRY IT 3.2

Evaluate:

.

216

EXAMPLE 4

.

Solution

In this expression, the variable is an exponent.

 Use the definition of exponent. Multiply.

When , the expression has a value of .

TRY IT 4.1

Evaluate:

.

64

TRY IT 4.2

Evaluate:

.

81

EXAMPLE 5

.

Solution

This expression contains two variables, so we must make two substitutions.

 Multiply. Add and subtract left to right.

When and , the expression has a value of .

TRY IT 5.1

Evaluate:

33

TRY IT 5.2

Evaluate:

10

EXAMPLE 6

.

Solution

We need to be careful when an expression has a variable with an exponent. In this expression, means and is different from the expression , which means .

TRY IT 6.1

Evaluate:

.

40

TRY IT 6.2

Evaluate:

.