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

$\frac{1+2}{3+4} = \frac{3}{7} \approx 0.4286$ .

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

$(1+2) \div (3+4) \approx 0.4286$ .

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, $\frac{1+ \frac{1}{2}}{1- \frac{1}{4}}$ , 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: $\frac{1+ \frac{1}{2}}{1- \frac{1}{4}}$ .

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Simplify the numerator: $1+\frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{2+1}{2} = \frac{3}{2}$ .

Simplify the denominator: $1- \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{4-1}{4} = \frac{3}{4}$ .

Divide: $\frac{1+ \frac{1}{2}}{1- \frac{1}{4}} = \frac{\frac{3}{2}}{\frac{3}{4}} = \frac{3}{2} \cdot \frac{4}{3} = \frac{3 \cdot 2 \cdot 2}{2 \cdot 3} = 2$ .

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

$(1+2÷2)÷(1-1÷4)=2$ .

Example

Evaluate: $\frac{\frac{2}{3} + \frac{1}{6}}{\frac{1}{2} + \frac{1}{4}}$ .

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Simplify the numerator: $\frac{2}{3} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{4+1}{6} = \frac{5}{6}$ .

Simplify the denominator: $\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{2+1}{4} = \frac{3}{4}$ .

Divide: $\frac{\frac{2}{3} + \frac{1}{6}}{\frac{1}{2} + \frac{1}{4}} = \frac{\frac{5}{6}}{\frac{3}{4}} = \frac{5}{6} \cdot \frac{4}{3} = \frac{5 \cdot 2 \cdot 2}{2 \cdot 3 \cdot 3} = \frac{5 \cdot 2}{3 \cdot 3} = \frac{10}{9} \approx 1.1111$ .

To evaluate the expression directly using a calculator,

$\frac{\frac{2}{3} + \frac{1}{6}}{\frac{1}{2} + \frac{1}{4}} = (2 \div 3+1 \div 4) \div (1 \div 2+1 \div 4) \approx 1.1111$ .

Example

Evaluate: $\frac{\frac{3^2}{5}}{\frac{2^2}{3}}$ .

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Simplify the numerator: $\frac{3^2}{5} = \frac{9}{5}$ .

Simplify the denominator: $\frac{2^2}{3} = \frac{3}{4}$ .

Divide: $\frac{\frac{9}{5}}{\frac{4}{3}} = \frac{9}{5} \cdot \frac{3}{4} = \frac{27}{20} = 1.35$ .

To evaluate the expression directly using a calculator, enter

$(3^2 \div 5) \div (2^2 \div 3) = 1.35$

Expressions Involving Fractions and Radicals

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

$\large \frac{\overline{x} - \mu _\overline{x}}{\sigma _ \overline{x}}$ ,

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

$\large \frac{\overline{x} - \mu _\overline{x}}{\frac{\sigma _x}{\sqrt{n}}}$ .

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.

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$\large \frac{\overline{x} - \mu _\overline{x}}{\frac{\sigma _x}{\sqrt{n}}} = \frac{15.8-16}{\frac{0.25}{\sqrt{36}}} = \frac{-0.2}{\frac{0.5}{6}} = \frac{-0.2}{1} \cdot \frac{6}{0.5} = \frac{-1.2}{0.5} = -2.4$

It is much faster to evaluate this expression using a calculator with one line entry.

$\large \frac{\overline{x} - \mu _\overline{x}}{\frac{\sigma _x}{\sqrt{n}}} =\frac{15.8-16}{\frac{0.25}{\sqrt{36}}} = (15.8-16) \div (0.25 \div \sqrt{36} ) = -2.4$

Note that many calculators insert a left parenthesis when you use the square root function. Therefore, your entry might look like this:

$\large \frac{15.8-16}{\frac{0.25}{\sqrt{36}}} =(15.8-16) \div (0.25 \div \sqrt{} (36) ) = -2.4$

Example

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

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Make sure you enter parentheses around the two terms in the numerator. The radical is an implied grouping.

$(4.5-5) \div \sqrt{(2^2 \div 25 +3^2 \div 16)} = -0.5882$

Example

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

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Make sure you enter parentheses around the two terms in the numerator. The radical is an implied grouping.

$(0.5-0.4) \div \sqrt{(0.45 \cdot (1-0.45) \cdot (1 \div 100 + 1 \div 100))} = 1.4213$

Module 10: Hypothesis Testing With Two Samples

Learning Outcomes

Complex Fractions

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Example

Example

Expressions Involving Fractions and Radicals

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Example

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Candela Citations