Evaluating Algebraic Expressions

Learning Outcomes

  • Use the order of operations to evaluate algebraic expressions

An algebraic expression is an expression involving addition, subtraction, multiplication, division, and exponentiation by an exponent that is a rational number. Radicals correspond to fractional exponents, so this definition means we can have integer exponents and radicals.

When you apply the order of operations to an expression involving a fraction, remember there are implied grouping symbols around the terms in the numerator and denominator.

For example, [latex]\frac{1+2}{3+4}[/latex] really means [latex]\frac{(1+2)}{(3+4)}[/latex]. Since we perform operations within grouping symbols first, [latex]\frac{1+2}{3+4} = \frac{3}{7}[/latex]. To find the decimal representation of this number using a calculator, you can simplify the numerator and denominator and then divide,

[latex]\frac{1+2}{3+4} = \frac{3}{7} \approx 0.4286[/latex].

Or, you can enter the original expression, inserting parentheses around the numerator and the denominator,

[latex](1+2) \div (3+4) \approx 0.4286[/latex].

In Module 10 you will encounter some complicated expressions involving fractions and radicals. In this section, we’ll review how to evaluate expressions like these.

Complex Fractions

A complex fraction is a fraction which contains fractions in the numerator and/or the denominator. For example, [latex]\frac{1+ \frac{1}{2}}{1- \frac{1}{4}}[/latex], is a complex fraction. The fractions in the numerator and denominator are called secondary fractions. To evaluate a complex fraction, you can simplify the numerator and denominator separately and then divide.

Example

Evaluate: [latex]\frac{1+ \frac{1}{2}}{1- \frac{1}{4}}[/latex].

To evaluate [latex]\frac{1+ \frac{1}{2}}{1- \frac{1}{4}}[/latex] on a calculator, as before, we insert parentheses around the numerator and denominator and divide:

[latex](1+2÷2)÷(1-1÷4)=2[/latex].

Example

Evaluate: [latex]\frac{\frac{2}{3} + \frac{1}{6}}{\frac{1}{2} + \frac{1}{4}}[/latex].

To evaluate the expression directly using a calculator,

[latex]\frac{\frac{2}{3} + \frac{1}{6}}{\frac{1}{2} + \frac{1}{4}} = (2 \div 3+1 \div 4) \div (1 \div 2+1 \div 4) \approx 1.1111[/latex].

Example

Evaluate: [latex]\frac{\frac{3^2}{5}}{\frac{2^2}{3}}[/latex].

To evaluate the expression directly using a calculator, enter

[latex](3^2 \div 5) \div (2^2 \div 3) = 1.35[/latex]

Expressions Involving Fractions and Radicals

When you found the test statistic for a hypothesis test for means,

[latex]\large \frac{\overline{x} - \mu _\overline{x}}{\sigma _ \overline{x}}[/latex],

the formula for the standard error was [latex]\sigma _\overline{x} = \frac{\sigma _x}{\sqrt{n}}[/latex]. So we could also have written the test statistics as

[latex]\large \frac{\overline{x} - \mu _\overline{x}}{\frac{\sigma _x}{\sqrt{n}}}[/latex].

This can be evaluated directly on a calculator as long as parentheses are inserted properly.

Example

Suppose that the weights of the contents of cereal boxes are normally distributed with a mean of 16 ounces and a standard deviation of 0.25 ounces. In a random sample of 36 boxes of cereal, the mean weight is observed to be 15.8 ounces. Find the value of the test statistic.

Example

Evaluate [latex]\large \frac{4.5-5}{\sqrt{\frac{2^2}{25} + \frac{3^2}{16}}}[/latex] using a calculator. Round your answer to four decimal places if necessary.

Example

Evaluate [latex]\large \frac{0.5-0.4}{\sqrt{0.45(1-0.45)(\frac{1}{100} + \frac{1}{100})}}[/latex] using a calculator.