A. Powers and Roots

Powers

A positive integer exponent or power is a convenient shorthand for indicating repeated multiplication of a number called the base. For example, the expression , read as “ squared,” means multiplied by . When either a positive or negative number is squared, the outcome will always be positive. When 0 is squared, the outcome is 0.

Example 1. Simplify .

The expression is read as “ cubed” and is read as “ to the th power,” or simply “ to the .” For positive integer powers of , it means that is multiplied times itself times.

Example 2. Simplify , and .

Notice that when a negative number is raised to an odd power, the result is negative, but when a negative number is raised to an even power, the result is positive.

Calculator Note: When raising negative numbers to powers, always be sure to enclose the negative number in parentheses before hitting the power key.

Example 3. Use a calculator to determine the square of .

On a TI-84, use the keystrokes . (Other calculators should be similar.) The result is (see the calculator screenshot below on the left).

Example 4. Repeat Example 3, but don’t enclose in parentheses.

On a TI-84, use the keystrokes , which gives the result (see the calculator screenshot above on the right). In this case (based on order of operations), 4 was squared first, resulting in 16, and then its opposite was taken, resulting in a final answer of .

Roots

The expression indicates a number that we square to get . The expression is called the “principal square root of ,” and the outcome is never negative. The expression Va, read as the “th root of ,” is the number that we raise to the th power to get .

Example 5. Determine and .

When a square root (or an th root) involves operations, it may be easier to simplify some of the calculations before taking the square root (or the th root).

Example 6. Approximate to one decimal place.

Example 7. The standard deviation of a set of numbers (introduced in Chapter 5 on page 204) involves a square root. Find the standard deviation of 2, , and 5. Give a simplified exact answer and an approximation to two decimal places.

The formula for computing the standard deviation of a set of numbers is

We follow the three steps from Algebra Review III, item A, Using Formulas (page AR-7).

Calculator Note: You can find a decimal approximation for the problem in Example 7 by entering the entire expression into your calculator before pressing the . The keystrokes for using a TI-84 to calculate the standard deviation of 2, , and 5 are given below.

After entering these keystrokes, your calculator screen should match the one below.

Warning: When entering expressions involving parentheses into a calculator, make sure every left parenthesis has a matching right parenthesis. Also, notice that some commands, such as the square root command, may automatically include a left parenthesis that you will need to pair with a right parenthesis.

Calculator Note: The TI-84 has a square root key but no th root key. Here’s how to compute the nth root of a number , , when :

Practice Exercises

B. natural and Fractional Exponents

We can extend the definition of exponent to 0 and negative integers as follows:

Example 1. Determine the values of , and .

Next, we extend the definition of exponent to fractions, in other words, exponents in the form with and integers and .

In defining (with if is even), it doesn’t matter if you raise to the mth power first or take the th root of first. You can do whichever is easier.

Example 2. Determine and .

or

In the computation of 8³, notice that it does not matter whether we took the cube root of 8 first and then squared the result or squared 8 first and then took the cube root.

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Example 3. Write the following radical expressions using exponents: and .

Calculator Note: When using a calculator, it may be easier to compute a constant raised to an exponent rather than the equivalent radical expression. See Example 4.

Example 4. Convert the following radical expressions to expressions involving exponents, and then use your calculator to approximate the values to two decimal places: and .

Here’s how these computations should appear on a TI-84 graphing calculator. Notice that the fractional exponents are enclosed in parentheses.

C. Graphs of Exponential Equations

Equations of the form are called exponential equations (or exponential functions). In such equations, the independent variable, , is an exponent. To understand the behavior of such a relation, we can sketch a graph by plotting points and then drawing a smooth curve through the plotted points.

Example 1. Determine points on the graph of , plot these points, and then draw a graph . Compare the graph of this exponential function with the graph of .

We begin by determining points on the graph of .

After plotting these points, we draw a smooth curve through them to obtain a graph of . Then we add a graph . The graphs of these two equations appear below.

By comparing the graphs of and , we can see that as the values of increase, the -values of the exponential equation increase faster than those of the linear equation. This will be true for exponential equations of the form , where .

Practice Exercises

D. Rules for Exponents and Roots

Rules for Exponents

When multiplying two exponential expressions with the same base, the rule is to add the exponents.

The statement above is easily understood when expanding the multiplication in a problem such as the following:

Example 1. Simplify and .

When dividing two exponential expressions with the same base, subtract the exponents.

This last statement is easily understood when simplifying a problem, such as , by cancelling as follows:

Example 2. Determine the value of and by cancellation and by using the rule for quotients.

When raising an exponential expression to a power, multiply powers.

To see how this rule works, we expand .

Example 3. Simplify both by expanding and by using the rule for raising to powers.

Warning: Don’t touch this! There is no “distributive law for exponents.” That is, in general,

Example 4. A student wrote the following equation on a test: . What is wrong with the student’s work? Correct the error.

The student tried to distribute the square power over the addition in . However, there is no distributive law for exponents. Instead the student should use the distributive law to expand the square.

Rules for Roots

The rules for roots may help us calculate a root by hand, as shown in Example 5.

Example 5. Evaluate and by hand.

The rules of roots can sometimes be used to simplify an expression containing a radical, as shown in Example 6.

Example 6. Simplify .

Warning: Don’t touch this! There is no rule that is the “distributive law for radicals.” That is, in general,

Example 7. A student wrote the following on a test: . What is wrong with the student’s work? Show the correct calculations.

There is no rule for addition of radicals. Instead, simplify the number under che radical sign first and then take the square root: .

Practice Exercises

In Practice Exercises 1–10, use the rules of exponents to simplify.

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In Practice Exercises 11 and 12, use the rules of roots (as shown in Example 5) to evaluate the root by hand.

E. Logarithms

A logarithm is an exponent, which represents the power that a number must be raised to in order to get another number: is the power that must be raised to in order to get . We read the expression as “the base- logarithm of .”

Example 1. Determine and .

because 3 must be raised to the second power in order to get 9.

because 3 must be raised to the fourth power in order to get 81.

Notice that a logarithmic equation has a corresponding exponential equation:

because

because

The subscript (or base) of the logarithmic equation is also the base of the corresponding exponential equation.

Because our number system is base 10, we often use a base-10 logarithm, which is also called a common logarithm. For common logarithms, the base is often omitted. On most calculators the key computes the base-10 logarithm of a number.

Example 2. Determine log(1000), log(10), and log(1).

The following two properties are true for logarithms when :

Example 3. Simplify for .

Another important base is , which is used in continuously compounding interest problems. Like the more familiar number , is an irrational number—that is, a nonrepeating, nonterminating decimal. Its value is approximately 2.718. The base-e logarithm could be written as . Mathematics, however, uses a special notation for base- logarithms, ln , and a special title, the natural logarithm.

Practice Exercises

For Practice Exercises 1–8, determine the value of the logarithm.

Simplify the expressions in Practice Exercises 9 and 10.

F. Using Logarithms to solve Equations

When an equation contains the variable of interest as an exponent, using logarithms can help solve the equation. In this section, we use the natural logarithm, the logarithm with a base of , ln . (Another approach would be to use the common logarithm, .)

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Example 1. Solve .

Because , we can estimate that the value of is between 3 and 4. Although we know the value of is closer to 3, we can’t state its exact value without logarithms. By taking the natural logarithm of both sides of the equation, we can get an exact solution. Here are the steps:

Using a calculator and rounding to two decimal places, we find that the approximate power is .

In the equation , when the variables , , and are stated, can be found by direct calculation. Solving for or is also straightforward (see Practice Exercises 3 and 4 in Algebra Review III, item B, Solving for One Variable in Terms of Another, page AR-9). However, if we want to solve for the exponent , we again turn to the technique of taking the natural logarithm of both sides of the equation.

Example 2. Solve for .

Practice Exercises

In Practice Exercises 1–3, find an exact solution, and then use your calculator to determine an approximate solution to two decimal places.