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Mathematics LibreTexts

9.5: Divide Square Roots

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Learning Objectives

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

  • Divide square roots
  • Rationalize a one-term denominator
  • Rationalize a two-term denominator
Note

Before you get started, take this readiness quiz.

  1. Find a fraction equivalent to 58 with denominator 48.
    If you missed this problem, review Exercise 1.6.1.
  2. Simplify: (5)2.
    If you missed this problem, review Example 9.4.13.
  3. Multiply: (7+3x)(7−3x).
    If you missed this problem, review Exercise 6.4.22.

Divide Square Roots

We know that we simplify fractions by removing factors common to the numerator and the denominator. When we have a fraction with a square root in the numerator, we first simplify the square root. Then we can look for common factors.

This figure shows two columns. The first is labeled “Common Factors” and has 3 times the square root of 2 over 3 times 5 beneath it. Both number threes are red. The second column is labeled “No common factors” and has 2 times the square root of 3 over 3 times 5.

Example 9.5.1

Simplify: 546

Answer
  546
Simplify the radical. 9·66
Simplify. 366
Remove the common factors. 363·2
Simplify. 62
Example 9.5.2

Simplify: 328.

Answer

22

Example 9.5.3

Simplify: 7515.

Answer

33

Example 9.5.4

Simplify: 62412.

Answer
  62412
Simplify the radical. 64·612
Simplify. 62612
Factor the common factor from the numerator. 2(36)12
Remove the common factors. 2(36)2·6
Simplify. 366
Example 9.5.5

Simplify: 84010.

Answer

4105

Example 9.5.6

Simplify: 107520.

Answer

534

We have used the Quotient Property of Square Roots to simplify square roots of fractions. The Quotient Property of Square Roots says

ab=ab, b0.

Sometimes we will need to use the Quotient Property of Square Roots ‘in reverse’ to simplify a fraction with square roots.

ab=ab, b0.

We will rewrite the Quotient Property of Square Roots so we see both ways together. Remember: we assume all variables are greater than or equal to zero so that their square roots are real numbers.

Definition: QUOTIENT PROPERTY OF SQUARE ROOTS

If a, b are non-negative real numbers and b0, then

ab=ab and ab=ab

We will use the Quotient Property of Square Roots ‘in reverse’ when the fraction we start with is the quotient of two square roots, and neither radicand is a perfect square. When we write the fraction in a single square root, we may find common factors in the numerator and denominator.

Example 9.5.7

Simplify: 2775

Answer
  2775
Neither radicand is a perfect square, so rewrite using the quotient property of square root. 2775
Remove common factors in the numerator and denominator. 925
Simplify. 35
Example 9.5.8

Simplify: 48108

Answer

23

Example 9.5.9

Simplify: 9654

Answer

43

We will use the Quotient Property for Exponents, aman=amn, when we have variables with exponents in the radicands.

Example 9.5.10

Simplify: 6y52y

Answer
  6y52y
Neither radicand is a perfect square, so rewrite using the quotient property of square root. 6y52y
Remove common factors in the numerator and denominator. 3y4
Simplify. y23
Example 9.5.11

Simplify: 12r36r.

Answer

r2

Example 9.5.12

Simplify: 14p92p5

Answer

p27

Example 9.5.13

Simplify: 72x3162x

Answer
  72x3162x
Rewrite using the quotient property of square roots. 72x3162x
Remove common factors. 18·4·x2·x18·9·x
Simplify. 4x29
Simplify the radical. 2x3
ExAMPLe 9.5.14

Simplify: 50s3128s.

Answer

5s8

Example 9.5.15

Simplify: 75q5108q.

Answer

5q26

Example 9.5.16

Simplify: 147ab83a3b4.

Answer
  147ab83a3b4
Rewrite using the quotient property of square roots. 147ab83a3b4
Remove common factors. \(\sqrt{\frac{49b^4}{a^2}}\
Simplify the radical. 7b2a
Example 9.5.17

Simplify: 162x10y22x6y6.

Answer

9x2y2

Example 9.5.18

Simplify: 300m3n73m5n.

Answer

10n3m

Rationalize a One Term Denominator

Before the calculator became a tool of everyday life, tables of square roots were used to find approximate values of square roots. Figure shows a portion of a table of squares and square roots. Square roots are approximated to five decimal places in this table.

This table has three solumn and eleven rows. The columns are labeled, “n,” “n squared,” and “the square root of n.” Under the column labeled “n” are the following numbers: 200; 201; 202; 203; 204; 205; 206; 207; 208; 209; and 210. Under the column labeled, “n squared” are the following numbers: 40,000; 40,401; 40,804; 41,209; 41,616; 42,025; 42,436; 42,849; 43,264; 43,681; 44,100. Under the column labeled, “the square root of n” are the following numbers: 14.14214; 14.17745; 14.21267; 14.24781; 14.28286; 14.31782; 14.35270; 14.38749; 14.42221; 14.45683; 14.49138.
A table of square roots was used to find approximate values of square roots before there were calculators.

If someone needed to approximate a fraction with a square root in the denominator, it meant doing long division with a five decimal-place divisor. This was a very cumbersome process.

For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. This process is still used today and is useful in other areas of mathematics, too.

Definition: RATIONALIZING THE DENOMINATOR

The process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer is called rationalizing the denominator.

Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator.

Let’s look at a numerical example.

Suppose we need an approximate value for the fraction.12A five decimal place approximation to2is1.4142111.41421Without a calculator, would you want to do this division?1.41421)¯1.0

But we can find a fraction equivalent to 12 by multiplying the numerator and denominator by 2.

This figure shows three fractions. The first fraction is 1 over the square root of 2. The second is 1 times the square root of 2 over the square root of 2 times the square root of 2. The third shows the square root of 2 over 2.

Now if we need an approximate value, we divide 2)¯1.41421. This is much easier.

Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a square root.

Similarly, a square root is not considered simplified if the radicand contains a fraction.

Definition: SIMPLIFIED SQUARE ROOTS

A square root is considered simplified if there are

  • no perfect-square factors in the radicand
  • no fractions in the radicand
  • no square roots in the denominator of a fraction

To rationalize a denominator, we use the property that (a)2=a. If we square an irrational square root, we get a rational number.

We will use this property to rationalize the denominator in the next example.

Example 9.5.19

Simplify: 43.

Answer

To remove the square root from the denominator, we multiply it by itself. To keep the fractions equivalent, we multiply both the numerator and denominator by the same factor.

  43

Multiply both the numerator and denominator by 3
4·33·3
Simplify. 433
Example 9.5.20

Simplify: 53.

Answer

533

Example 9.5.21

Simplify: 65.

Answer

655

Example 9.5.22

Simplify: 836

Answer

To remove the square root from the denominator, we multiply it by itself. To keep the fractions equivalent, we multiply both the numerator and denominator by 6.

  .
Multiply both the numerator and the denominator by 6. .
Simplify. .
Remove common factors. .
Simplify. .
Example 9.5.23

Simplify: 525.

Answer

52

Example 9.5.24

Simplify: 943.

Answer

334

Always simplify the radical in the denominator first, before you rationalize it. This way the numbers stay smaller and easier to work with.

Example 9.5.25

Simplify: 512.

Answer
  .
The fraction is not a perfect square, so rewrite using the
Quotient Property.
.
Simplify the denominator. .
Rationalize the denominator. .
Simplify. .
Simplify. .
Example 9.5.26

Simplify: 718.

Answer

146

Example 9.5.27

Simplify: 332.

Answer

68

Example 9.5.28

Simplify: 1128

Answer
  .
Rewrite using the Quotient Property. .
Simplify the denominator. .
Rationalize the denominator. .
Simplify. .
Simplify. .
Example 9.5.29

Simplify: 327.

Answer

13

Example 9.5.30

Simplify: 1050

Answer

55

Rationalize a Two-Term Denominator

When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates pattern to rationalize the denominator.

(ab)(a+b)(25)(2+5)a2b222(5)2451

When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.

Example 9.5.31

Simplify: 44+2.

Answer
  .
Multiply the numerator and denominator by the conjugate of the denominator. .
Multiply the conjugates in the denominator. .
Simplify the denominator. .
Simplify the denominator. .
Remove common factors from the numerator and denominator. .
We leave the numerator in factored form to make it easier to look for common factors after we have simplified the denominator.  
Example 9.5.32

Simplify: 22+3.

Answer

2(23)1

Example 9.5.33

Simplify: 55+3.

Answer

5(53)22

Example 9.5.34

Simplify: 523.

Answer
  .
Multiply the numerator and denominator by the conjugate of the denominator. .
Multiply the conjugates in the denominator. .
Simplify the denominator. .
Simplify the denominator. .
Simplify. .
Example 9.5.35

Simplify: 315.

Answer

3(1+5)4

Example 9.5.36

Simplify: 246.

Answer

4+65

Example 9.5.37

Simplify: 3u6.

Answer
  .
Multiply the numerator and denominator by the conjugate of the denominator. .
Multiply the conjugates in the denominator. .
Simplify the denominator. .
Example 9.5.38

Simplify: 5x+2.

Answer

5(x2)x2

Example 9.5.39

Simplify: 10y3.

Answer

10(y+3)y3

Example 9.5.40

Simplify:x+7x7.

Answer
  .
Multiply the numerator and denominator by the conjugate of the denominator. .
Multiply the conjugates in the denominator. .
Simplify the denominator. .
We do not square the numerator. In factored form, we can see there are no common factors to remove from the numerator and denominator.  
Example 9.5.41

Simplify: p+2p2.

Answer

(p+2)2p2

Example 9.5.42

Simplify: q10q+10.

Answer

(q10)2q10

Access this online resource for additional instruction and practice with dividing and rationalizing.

  • Dividing and Rationalizing

Key Concepts

  • Quotient Property of Square Roots
    • If a, b are non-negative real numbers and b0, then

      ab=ab and ab=ab

  • Simplified Square Roots
    A square root is considered simplified if there are
    • no perfect square factors in the radicand
    • no fractions in the radicand
    • no square roots in the denominator of a fraction

Glossary

rationalizing the denominator
The process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer is called rationalizing the denominator.

This page titled 9.5: Divide Square Roots is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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