7.5 Simplify Square Roots – Introductory Algebra (2024)

Table of Contents
Use the Product Property to Simplify Square Roots Use the Quotient Property to Simplify Square Roots Everyday Math Writing Exercises FAQs

CHAPTER 7 Powers, Roots, and Scientific Notation

Learning Objectives

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

  • Use the Product Property to simplify square roots
  • Use the Quotient Property to simplify square roots

In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that 7.5 Simplify Square Roots – Introductory Algebra (1) is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use in (Simplify and Use Square Roots).

But what if we want to estimate 7.5 Simplify Square Roots – Introductory Algebra (2)? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons, too, to simplify square roots as you’ll see later in this chapter.

A square root is considered simplified if its radicand contains no perfect square factors.

Simplified Square Root

7.5 Simplify Square Roots – Introductory Algebra (3) is considered simplified if 7.5 Simplify Square Roots – Introductory Algebra (4) has no perfect square factors.

So 7.5 Simplify Square Roots – Introductory Algebra (5) is simplified. But 7.5 Simplify Square Roots – Introductory Algebra (6) is not simplified, because 16 is a perfect square factor of 32

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that 7.5 Simplify Square Roots – Introductory Algebra (7). The corresponding property of square roots says that 7.5 Simplify Square Roots – Introductory Algebra (8).

Product Property of Square Roots

If a, b are non-negative real numbers, then 7.5 Simplify Square Roots – Introductory Algebra (9).

We use the Product Property of Square Roots to remove all perfect square factors from a radical. We will show how to do this in (Example 1).

EXAMPLE 1

How To Use the Product Property to Simplify a Square Root

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (10).

Solution

7.5 Simplify Square Roots – Introductory Algebra (11)7.5 Simplify Square Roots – Introductory Algebra (12)7.5 Simplify Square Roots – Introductory Algebra (13)

TRY IT 1.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (14).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (15)

TRY IT 1.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (16).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (17)

Notice in the previous example that the simplified form of 7.5 Simplify Square Roots – Introductory Algebra (18) is 7.5 Simplify Square Roots – Introductory Algebra (19), which is the product of an integer and a square root. We always write the integer in front of the square root.

HOW TO: Simplify a square root using the product property.
  1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
  2. Use the product rule to rewrite the radical as the product of two radicals.
  3. Simplify the square root of the perfect square.

EXAMPLE 2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (20).

Solution

7.5 Simplify Square Roots – Introductory Algebra (21)
Rewrite the radicand as a product using the largest perfect square factor.7.5 Simplify Square Roots – Introductory Algebra (22)
Rewrite the radical as the product of two radicals.7.5 Simplify Square Roots – Introductory Algebra (23)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (24)

TRY IT 2.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (25).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (26)

TRY IT 2.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (27).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (28)

We could use the simplified form 7.5 Simplify Square Roots – Introductory Algebra (29) to estimate 7.5 Simplify Square Roots – Introductory Algebra (30). We know 5 is between 2 and 3, and 7.5 Simplify Square Roots – Introductory Algebra (31) is 7.5 Simplify Square Roots – Introductory Algebra (32). So 7.5 Simplify Square Roots – Introductory Algebra (33) is between 20 and 30.

The next example is much like the previous examples, but with variables.

EXAMPLE 3

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (34).

Solution

7.5 Simplify Square Roots – Introductory Algebra (35)
Rewrite the radicand as a product using the largest perfect square factor.7.5 Simplify Square Roots – Introductory Algebra (36)
Rewrite the radical as the product of two radicals.7.5 Simplify Square Roots – Introductory Algebra (37)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (38)

TRY IT 3.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (39).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (40)

TRY IT 3.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (41).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (42)

We follow the same procedure when there is a coefficient in the radical, too.

EXAMPLE 4

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (43)

Solution

7.5 Simplify Square Roots – Introductory Algebra (44)
Rewrite the radicand as a product using the largest perfect square factor.7.5 Simplify Square Roots – Introductory Algebra (45)
Rewrite the radical as the product of two radicals.7.5 Simplify Square Roots – Introductory Algebra (46)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (47)

TRY IT 4.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (50).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (51)

In the next example both the constant and the variable have perfect square factors.

EXAMPLE 5

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (52).

Solution

7.5 Simplify Square Roots – Introductory Algebra (53)
Rewrite the radicand as a product using the largest perfect square factor.7.5 Simplify Square Roots – Introductory Algebra (54)
Rewrite the radical as the product of two radicals.7.5 Simplify Square Roots – Introductory Algebra (55)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (56)

TRY IT 5.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (57).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (58)

TRY IT 5.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (59).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (60)

EXAMPLE 6

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (61).

Solution

7.5 Simplify Square Roots – Introductory Algebra (62)
Rewrite the radicand as a product using the largest perfect square factor.7.5 Simplify Square Roots – Introductory Algebra (63)
Rewrite the radical as the product of two radicals.7.5 Simplify Square Roots – Introductory Algebra (64)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (65)

TRY IT 6.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (66).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (67)

TRY IT 6.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (68).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (69)

We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify 7.5 Simplify Square Roots – Introductory Algebra (70) we must simplify each square root separately first, then add to get the sum of 17

The expression 7.5 Simplify Square Roots – Introductory Algebra (71) cannot be simplified—to begin we’d need to simplify each square root, but neither 17 nor 7 contains a perfect square factor.

In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer.

EXAMPLE 7

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (72).

Solution

7.5 Simplify Square Roots – Introductory Algebra (73)
Rewrite the radicand as a product using the largest perfect square factor.7.5 Simplify Square Roots – Introductory Algebra (74)
Rewrite the radical as the product of two radicals.7.5 Simplify Square Roots – Introductory Algebra (75)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (76)

The terms are not like and so we cannot add them. Trying to add an integer and a radical is like trying to add an integer and a variable—they are not like terms!

TRY IT 7.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (77).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (78)

TRY IT 7.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (79).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (80)

The next example includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.

EXAMPLE 8

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (81).

Solution

7.5 Simplify Square Roots – Introductory Algebra (82)
Rewrite the radicand as a product using the largest perfect square factor.7.5 Simplify Square Roots – Introductory Algebra (83)
Rewrite the radical as the product of two radicals.7.5 Simplify Square Roots – Introductory Algebra (84)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (85)
Factor the common factor from the numerator.7.5 Simplify Square Roots – Introductory Algebra (86)
Remove the common factor, 2, from the numerator and denominator.7.5 Simplify Square Roots – Introductory Algebra (87)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (88)

TRY IT 8.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (89).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (90)

TRY IT 8.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (91).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (92)

Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.

TRY IT 9.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (97).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (98)

TRY IT 9.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (99).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (100)

If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!

EXAMPLE 10

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (101).

Solution

7.5 Simplify Square Roots – Introductory Algebra (102)
Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.7.5 Simplify Square Roots – Introductory Algebra (103)
Simplify the fraction by removing common factors.7.5 Simplify Square Roots – Introductory Algebra (104)
7.5 Simplify Square Roots – Introductory Algebra (105)7.5 Simplify Square Roots – Introductory Algebra (106)

TRY IT 10.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (107).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (108)

TRY IT 10.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (109).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (110)

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, 7.5 Simplify Square Roots – Introductory Algebra (111).

EXAMPLE 11

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (112).

Solution

7.5 Simplify Square Roots – Introductory Algebra (113)
Simplify the fraction inside the radical first. Divide the like bases by subtracting the exponents.7.5 Simplify Square Roots – Introductory Algebra (114)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (115)

TRY IT 11.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (116).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (117)

TRY IT 11.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (118).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (119)

EXAMPLE 12

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (120).

Solution

7.5 Simplify Square Roots – Introductory Algebra (121)
Simplify the fraction inside the radical first.7.5 Simplify Square Roots – Introductory Algebra (122)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (123)

TRY IT 12.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (124).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (125)

TRY IT 12.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (126).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (127)

Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.

7.5 Simplify Square Roots – Introductory Algebra (128)

We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square we simplify the numerator and denominator separately.

Quotient Property of Square Roots

If a, b are non-negative real numbers and 7.5 Simplify Square Roots – Introductory Algebra (129), then

7.5 Simplify Square Roots – Introductory Algebra (130)

EXAMPLE 13

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (131).

Solution

7.5 Simplify Square Roots – Introductory Algebra (132)
We cannot simplify the fraction inside the radical. Rewrite using the quotient property.7.5 Simplify Square Roots – Introductory Algebra (133)
Simplify the square root of 64. The numerator cannot be simplified.7.5 Simplify Square Roots – Introductory Algebra (134)

TRY IT 13.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (135).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (136)

TRY IT 13.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (137).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (138)

EXAMPLE 14

How to Use the Quotient Property to Simplify a Square Root

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (139).

Solution

7.5 Simplify Square Roots – Introductory Algebra (140)7.5 Simplify Square Roots – Introductory Algebra (141)7.5 Simplify Square Roots – Introductory Algebra (142)

TRY IT 14.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (143).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (144)

TRY IT 14.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (145).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (146)

HOW TO: Simplify a square root using the quotient property.
  1. Simplify the fraction in the radicand, if possible.
  2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
  3. Simplify the radicals in the numerator and the denominator.

EXAMPLE 15

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (147).

Solution

7.5 Simplify Square Roots – Introductory Algebra (148)
We cannot simplify the fraction in the radicand. Rewrite using the Quotient Property.7.5 Simplify Square Roots – Introductory Algebra (149)
Simplify the radicals in the numerator and the denominator.7.5 Simplify Square Roots – Introductory Algebra (150)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (151)

TRY IT 15.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (152).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (153)

TRY IT 15.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (154).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (155)

Be sure to simplify the fraction in the radicand first, if possible.

EXAMPLE 16

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (156).

Solution

7.5 Simplify Square Roots – Introductory Algebra (157)
Simplify the fraction in the radicand.7.5 Simplify Square Roots – Introductory Algebra (158)
Rewrite using the Quotient Property.7.5 Simplify Square Roots – Introductory Algebra (159)
Simplify the radicals in the numerator and the denominator.7.5 Simplify Square Roots – Introductory Algebra (160)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (161)

TRY IT 16.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (162).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (163)

TRY IT 16.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (164).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (165)

EXAMPLE 17

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (166).

Solution

7.5 Simplify Square Roots – Introductory Algebra (167)
Simplify the fraction in the radicand, if possible.7.5 Simplify Square Roots – Introductory Algebra (168)
Rewrite using the Quotient Property.7.5 Simplify Square Roots – Introductory Algebra (169)
Simplify the radicals in the numerator and the denominator.7.5 Simplify Square Roots – Introductory Algebra (170)
Simplify.7.5 Simplify Square Roots – Introductory Algebra (171)

TRY IT 17.1

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (172).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (173)

TRY IT 17.2

Simplify: 7.5 Simplify Square Roots – Introductory Algebra (174).

Show answer

7.5 Simplify Square Roots – Introductory Algebra (175)

  • Simplified Square Root7.5 Simplify Square Roots – Introductory Algebra (176) is considered simplified if 7.5 Simplify Square Roots – Introductory Algebra (177) has no perfect-square factors.
  • Product Property of Square Roots If a, b are non-negative real numbers, then

    7.5 Simplify Square Roots – Introductory Algebra (178)

  • Simplify a Square Root Using the Product Property To simplify a square root using the Product Property:
    1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.
    2. Use the product rule to rewrite the radical as the product of two radicals.
    3. Simplify the square root of the perfect square.
  • Quotient Property of Square Roots If a, b are non-negative real numbers and 7.5 Simplify Square Roots – Introductory Algebra (179), then

    7.5 Simplify Square Roots – Introductory Algebra (180)

  • Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property:
    1. Simplify the fraction in the radicand, if possible.
    2. Use the Quotient Rule to rewrite the radical as the quotient of two radicals.
    3. Simplify the radicals in the numerator and the denominator.

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

1. 7.5 Simplify Square Roots – Introductory Algebra (181)2. 7.5 Simplify Square Roots – Introductory Algebra (182)
3. 7.5 Simplify Square Roots – Introductory Algebra (183)4. 7.5 Simplify Square Roots – Introductory Algebra (184)
5. 7.5 Simplify Square Roots – Introductory Algebra (185)6. 7.5 Simplify Square Roots – Introductory Algebra (186)
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47. 7.5 Simplify Square Roots – Introductory Algebra (227)48. 7.5 Simplify Square Roots – Introductory Algebra (228)


Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

49. 7.5 Simplify Square Roots – Introductory Algebra (229)50. 7.5 Simplify Square Roots – Introductory Algebra (230)
51. 7.5 Simplify Square Roots – Introductory Algebra (231)52. 7.5 Simplify Square Roots – Introductory Algebra (232)
53. 7.5 Simplify Square Roots – Introductory Algebra (233)54. 7.5 Simplify Square Roots – Introductory Algebra (234)
55. 7.5 Simplify Square Roots – Introductory Algebra (235)56. 7.5 Simplify Square Roots – Introductory Algebra (236)
57. 7.5 Simplify Square Roots – Introductory Algebra (237)58. 7.5 Simplify Square Roots – Introductory Algebra (238)
59. 7.5 Simplify Square Roots – Introductory Algebra (239)60. 7.5 Simplify Square Roots – Introductory Algebra (240)
61. 7.5 Simplify Square Roots – Introductory Algebra (241)62. 7.5 Simplify Square Roots – Introductory Algebra (242)
63. 7.5 Simplify Square Roots – Introductory Algebra (243)64. 7.5 Simplify Square Roots – Introductory Algebra (244)
65. 7.5 Simplify Square Roots – Introductory Algebra (245)66. 7.5 Simplify Square Roots – Introductory Algebra (246)
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87. 7.5 Simplify Square Roots – Introductory Algebra (267)88. 7.5 Simplify Square Roots – Introductory Algebra (268)

Everyday Math

89.

a) Elliott decides to construct a square garden that will take up 288 square feet of his yard. Simplify 7.5 Simplify Square Roots – Introductory Algebra (269) to determine the length and the width of his garden. Round to the nearest tenth of a foot.

b) Suppose Elliott decides to reduce the size of his square garden so that he can create a 5-foot-wide walking path on the north and east sides of the garden. Simplify 7.5 Simplify Square Roots – Introductory Algebra (270) to determine the length and width of the new garden. Round to the nearest tenth of a foot.

90.

a) Melissa accidentally drops a pair of sunglasses from the top of a roller coaster, 64 feet above the ground. Simplify 7.5 Simplify Square Roots – Introductory Algebra (271) to determine the number of seconds it takes for the sunglasses to reach the ground.

b) Suppose the sunglasses in the previous example were dropped from a height of 144 feet. Simplify 7.5 Simplify Square Roots – Introductory Algebra (272) to determine the number of seconds it takes for the sunglasses to reach the ground.

Writing Exercises

91. Explain why 7.5 Simplify Square Roots – Introductory Algebra (273). Then explain why 7.5 Simplify Square Roots – Introductory Algebra (274).92. Explain why 7.5 Simplify Square Roots – Introductory Algebra (275) is not equal to 7.5 Simplify Square Roots – Introductory Algebra (276).
1. 7.5 Simplify Square Roots – Introductory Algebra (277)3. 7.5 Simplify Square Roots – Introductory Algebra (278)5. 7.5 Simplify Square Roots – Introductory Algebra (279)
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61. 7.5 Simplify Square Roots – Introductory Algebra (307)63. 7.5 Simplify Square Roots – Introductory Algebra (308)65. 7.5 Simplify Square Roots – Introductory Algebra (309)
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73. 7.5 Simplify Square Roots – Introductory Algebra (313)75. 7.5 Simplify Square Roots – Introductory Algebra (314)77. 7.5 Simplify Square Roots – Introductory Algebra (315)
79. 7.5 Simplify Square Roots – Introductory Algebra (316)81. 7.5 Simplify Square Roots – Introductory Algebra (317)83. 7.5 Simplify Square Roots – Introductory Algebra (318)
85. 7.5 Simplify Square Roots – Introductory Algebra (319)87. 7.5 Simplify Square Roots – Introductory Algebra (320)89. a)7.5 Simplify Square Roots – Introductory Algebra (321)b)7.5 Simplify Square Roots – Introductory Algebra (322)
91.Answers will vary.

This chapter has been adapted from “Simplify Square Roots” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

7.5 Simplify Square Roots – Introductory Algebra (2024)

FAQs

How do you simplify square root answers? ›

Step 1: Find the prime factors of the number inside the radical sign. Step 2: Group the factors into pairs. Step 3: Pull out one integer outside the radical sign for each pair. Leave the other integers that could not be paired inside the radical sign.

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How to simplify the square root of 7? ›

√7 cannot be simplified any further as it is prime. The radical form of the square root of 7 is √7.

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What is a square root of 75? ›

We know that the square root of 75 is 8.66.

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How to simplify a square root with a variable? ›

1) If the expression consists of a variable raised to an even power, the square root of the expression equals the variable raised to one-half of that power. Examples: 2) If the variable contains an odd power, express it as the product of two factors, one having an exponent 1 and the other with an even exponent.

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What is sqrt 75 in simplest form? ›

In this case (√25 x 3) would be another way of writing √75. Separate the surd, √25 x √3. The √25 is 5 so this can now be written as 5√3. As √3 can not be simplified any further, the surd is now written in its simplest form.

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How do you simplify a root expression? ›

Simplifying Square Roots Without Fractions

If a factor of the radicand contains a variable with an odd exponent, the square root is obtained by first factoring the variable factor into two factors so that one has an even exponent and the other has an exponent of 1, then using the product property of square roots.

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How to tell if a square root can be simplified? ›

You can simplify a radical if it has a factor of the appropriate power — sticking with square roots for the time being, you can simplify a square root if it has a square factor. √50, for example, is √(2 x 5 x 5), so you can rewrite that as 5√2.

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How to simplify the square root of 145? ›

sqrt(145) No simplification exists, Root remains : • sqrt(145) Simplify : sqrt(145) Factor 145 into its prime factors 145 = 5 • 29 To simplify a square root, we extract ...

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