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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

Before you get started, take this readiness quiz.

  1. Simplify: 9 2 ( −9 ) 2 9 2 .
    If you missed this problem, review [link] .
  2. Round 3.846 to the nearest hundredth.
    If you missed this problem, review [link] .
  3. For each number, identify whether it is a real number or not a real number:
    100 −100 .
    If you missed this problem, review [link] .

Simplify expressions with square roots

Remember that when a number n is multiplied by itself, we write n 2 and read it “n squared.” For example, 15 2 reads as “15 squared,” and 225 is called the square of 15, since 15 2 = 225 .

Square of a number

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

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 225 is the square of 15, we can also say that 15 is a square root of 225. 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 ( −15 ) 2 = 225 also, so −15 is also a square root of 225. Therefore, both 15 and −15 are square roots of 225.

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 also called 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

This figure is a picture of an m inside a square root sign. The sign is labeled as a radical sign and the m is labeled as the radicand.

m is read as “the square root of m .”

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 15 is the positive square root of 225, we write 225 = 15 . Fill in [link] to make a table of square roots you can refer to as you work this chapter.

This table has fifteen columns and two rows. The first row contains the following numbers: the square root of 1, the square root of 4, the square root of 9, the square root of 16, the square root of 25, the square root of 36, the square root of 49, the square root of 64, the square root of 81, the square root of 100, the square root of 121, the square root of 144, the square root of 169, the square root of 196, and the square root of 225. The second row is completely empty except for the last column. The number 15 is in the last column.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write 225 = 15 . If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, 225 = −15 .

Simplify: 36 196 81 289 .

Solution


36 Since 6 2 = 36 6


196 Since 14 2 = 196 14


81 The negative is in front of the radical sign. −9


289 The negative is in front of the radical sign. −17

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Simplify: 49 225 .

−7 15

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Simplify: 64 121 .

8 −11

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Simplify: −169 64 .

Solution


  1. −169 There is no real number whose square is −169 . −169 is not a real number.


  2. 64 The negative is in front of the radical. −8
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Simplify: −196 81 .

not a real number −9

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Simplify: 49 −121 .

−7 not a real number

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When using the order of operations to simplify an expression that has square roots, we treat the radical as a grouping symbol.

Simplify: 25 + 144 25 + 144 .

Solution


25 + 144 Use the order of operations. 5 + 12 Simplify. 17


25 + 144 Simplify under the radical sign. 169 Simplify. 13

Notice the different answers in parts and !

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Simplify: 9 + 16 9 + 16 .

7 5

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Simplify: 64 + 225 64 + 225 .

17 23

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Estimate square roots

So far we have only considered square roots of perfect square numbers. The square roots of other numbers are not whole numbers. Look at [link] below.

Practice Key Terms 3

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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