5.7 Simplify and Use Square Roots

Learning Objectives

By the end of this section, you will be able to:

  • Simplify expressions with square roots
  • Estimate square roots
  • Approximate square roots
  • Simplify variable expressions with square roots
  • Use square roots in applications

Simplify Expressions with Square Roots

To start this section, we need to review some important vocabulary and notation.

Remember that when a number n is multiplied by itself, we can write this as n2, which we read aloud as “nsquared.” For example, 82 is read as “8squared.”

We call 64 the square of 8 because 82=64. Similarly, 121 is the square of 11, because 112=121.

Square Root of a Negative Number

Can we simplify √ −25? Is there a number whose square is −25?

None of the numbers that we have dealt with so far have a square that is −25. Why? Any positive number squared is positive, and any negative number squared is also positive. In the next chapter we will see that all the numbers we work with are called the real numbers. So we say there is no real number equal to √ −25. If we are asked to find the square root of any negative number, we say that the solution is not a real number.

Square Roots and the Order of Operations

When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.

Notice the different answers in parts ⓐ and ⓑ of Example 5.72. It is important to follow the order of operations correctly. In ⓐ, we took each square root first and then added them. In ⓑ, we added under the radical sign first and then found the square root.

Estimate Square Roots

So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.

A table is shown with 2 columns. The first column is labeled “Number” and contains the values: 4, 5, 6, 7, 8, 9. The second column is labeled “Square root” and contains the values: square root of 4 equals 2, square root of 5, square root of 6, square root of 7, square root of 8, square root of 9 equals 3.

We might conclude that the square roots of numbers between 44 and 99 will be between 2 and 3, and they will not be whole numbers.

Based on the pattern in the table above, we could say that √5 is between 2 and 3. Using inequality symbols, we write


Approximate Square Roots with a Calculator

There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots.

Find the –√ or x−−√ key on your calculator. You will to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator’s display. The symbol for an approximation is ≈ and it is read approximately.

Suppose your calculator has a 10-digit display. Using it to find the square root of 5 will give 2.236067977. This is the approximate square root of 5. When we report the answer, we should use the “approximately equal to” sign instead of an equal sign.

You will seldom use this many digits for applications in algebra. So, if you wanted to round 5–√ to two decimal places, you would write

How do we know these values are approximations and not the exact values? Look at what happens when we square them.

The squares are close, but not exactly equal, to 5.

Use Square Roots in Applications

As you progress through your college courses, you’ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer!

Square Roots and Area

We have solved applications with area before. If we were given the length of the sides of a square, we could find its area by squaring the length of its sides. Now we can find the length of the sides of a square if we are given the area, by finding the square root of the area.

If the area of the square is AA square units, the length of a side is A−−√A units. See Table 5.7.

Area (square units)Length of side (units)


Square Roots and Accident Investigations

Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is dd feet, then the speed of the car can be found by evaluating √24d.

Section 5.7 Exercises

Practice Makes Perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.


ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?