<< Chapter < Page Chapter >> Page >
By the end of this section, you will be able to:
  • Simplify expressions with square roots
  • Identify integers, rational numbers, irrational numbers, and real numbers
  • Locate fractions on the number line
  • Locate decimals on the number line

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapters, Decimals and Properties of Real Numbers .

Simplify expressions with square roots

Remember that when a number n is multiplied by itself, we write n 2 and read it “n squared.” The result is called the square of n . For example,

8 2 read 8 squared’ 64 64 is called the square of 8 .

Similarly, 121 is the square of 11, because 11 2 is 121.

Square of a number

If n 2 = m , then m is the square of n .

Doing the Manipulative Mathematics activity “Square Numbers” will help you develop a better understanding of perfect square numbers.

Complete the following table to show the squares of the counting numbers 1 through 15.

There is a table with two rows and 17 columns. The first row reads from left to right Number, n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second row reads from left to right Square, n squared, blank, blank, blank, blank, blank, blank, blank, 64, blank, blank, 121, blank, blank, blank, and blank.

The numbers in the second row are called perfect square numbers. It will be helpful to learn to recognize the perfect square numbers.

The squares of the counting numbers are positive numbers. What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.

( −3 ) 2 = 9 ( −8 ) 2 = 64 ( −11 ) 2 = 121 ( −15 ) 2 = 225

Did you notice that these squares are the same as the squares of the positive numbers?

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 10 2 = 100 , we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is m is called a square root of m .

Square root of a number

If n 2 = m , then n is a square root of m .

Notice ( −10 ) 2 = 100 also, so −10 is also a square root of 100. Therefore, both 10 and −10 are square roots of 100.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign    , m , denotes the positive square root. The positive square root is called the principal square root . When we use the radical sign that always means we want the principal square root.

We also use the radical sign for the square root of zero. Because 0 2 = 0 , 0 = 0 . Notice that zero has only one square root.

Square root notation

m is read “the square root of m

A square root is given, with an arrow to the radical sign (it looks like a checkmark with a horizontal line extending from its long end) denoted radical sign and an arrow to the number under the radical sign, which is marked radicand.

If m = n 2 , then m = n , for n 0 .

The square root of m , m , is the positive number whose square is m .

Since 10 is the principal square root of 100, we write 100 = 10 . You may want to complete the following table to help you recognize square roots.

There is a table with two rows and 15 columns. The first row reads from left to right square root of 1, square root of 4, square root of 9, square root of 16, square root of 25, square root of 36, square root of 49, square root of 64, square root of 81, square root of 100, square root of 121, square root of 144, square root of 169, square root of 196, and square root of 225. The second row consists of all blanks except for the tenth cell under the square root of 100, which reads 10.

Simplify: 25 121 .

Solution


25 Since 5 2 = 25 5


121 Since 11 2 = 121 11

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Simplify: 36 169 .

6 13

Got questions? Get instant answers now!

Simplify: 16 196 .

4 14

Got questions? Get instant answers now!

We know that every positive number has two square roots and the radical sign indicates the positive one. We write 100 = 10 . If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, 100 = −10 . We read 100 as “the opposite of the square root of 10.”

Simplify: 9 144 .


  1. 9 The negative is in front of the radical sign. 3

  2. 144 The negative is in front of the radical sign. 12
Got questions? Get instant answers now!
Got questions? Get instant answers now!
Practice Key Terms 6

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask