2.3 Entire and Mixed Radicals

An entire radical is a radical with a coefficient equal to 1.

Examples of entire radicals: \sqrt{25}    \sqrt{84}    \sqrt[3]{18}    \sqrt[5]{-32}

A mixed radical is a radical with a coefficient equal to something other than 1.

Examples of mixed radicals: 2\sqrt{15}    \frac{1}{4}\sqrt{64}    -5\sqrt[3]{29}    6\sqrt[4]{55}

Converting Entire Radicals (index = 2) into Mixed Radicals

An entire radical can often be converted into a mixed radical. For example, \sqrt{50} = 5\sqrt{2}.

Table 2.3.1
To convert an entire radical with an index of 2 into a mixed radical:
Step 1: Find the largest factor in the radicand that is a perfect square. Rewrite the radicand as a product of the perfect square and another number.

Step 2: Use the product rule (\sqrt{ab} = \sqrt{a} × \sqrt{b} where a, b \geq 0) to rewrite the radical as the product of two radicals.

Step 3: Simplify the square root of the perfect square. A mixed radical with an index of 2 is considered simplified once all perfect squares have been factored out of the radicand and square rooted.
Example 1

Convert \sqrt{50} into a mixed radical.

Table 2.3.2
Steps Solution
Find the largest factor in the radicand that is a perfect square. Rewrite the radicand as a product of the perfect square and another number. \sqrt{50}

\sqrt{25 \bullet 2}
Use the product rule (\sqrt{ab} = \sqrt{a} × \sqrt{b} where a, b \geq 0) to rewrite the radical as the product of two radicals. \sqrt{25} \times \sqrt{2}
Simplify the square root of the perfect square. 5 \times \sqrt{2}

5\sqrt{2}
Practice 1

Atya was asked to convert \sqrt{48} into a mixed radical in simplest form. She used the rule \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} to start her work. Complete her work.

\sqrt{48}

= \sqrt{16 \times 3}

= \sqrt{16} \times \sqrt{3}

 
Practice 2

Convert the following entire radicals into mixed radicals in simplest form.

a) \sqrt{45}

 

 

 b) \sqrt{80}

 

 

 c) \sqrt{343}

 

 
Practice 3

Convert the following radicals into mixed radicals in simplest form.

a) 4\sqrt{50}

 

 

 

 b) 3\sqrt{240}

 

 

 

 c) \frac{2}{3}\sqrt{45}

 

 

 

Converting Mixed Radicals (index = 2) into Entire Radicals

Every mixed radical can be converted into an entire radical. For example, 5\sqrt{2} = \sqrt{50}.

Table 2.3.4
To convert a mixed radical with an index of 2 into an entire radical:
Step 1: Write your coefficient as a radical by squaring the coefficient and writing this number under a radical sign.

Step 2: Use the product rule (\sqrt{ab} = \sqrt{a} × \sqrt{b} where a, b \geq 0) to rewrite the two radicals as a single radical.

Step 3: Multiply the two numbers under the radical sign to simplify.
Example 2

Convert 5\sqrt{2} into an entire radical.

Table 2.3.5
Steps Solution
Write your coefficient as a radical by squaring the coefficient and writing this number under a radical sign. 5\sqrt{2}

5 \times \sqrt{2}

\sqrt{25} \times \sqrt{2}
Use the product rule (\sqrt{ab} = \sqrt{a} × \sqrt{b} where a, b \geq 0) to rewrite the two radicals as a single radical. \sqrt{25 \bullet 2}
Multiply the two numbers under the radical sign to simplify. \sqrt{50}
Practice 4

Convert the following mixed radicals into entire radicals.

a) 4\sqrt{3}

 

 

 

 b) 6\sqrt{5}

 

 

 

 c) -7\sqrt{7}

 

 

 

 d) 0.25\sqrt{48} 

 

 

 

 e) \frac{3}{4}\sqrt{32}  

 

 

 
Practice 5

A square backyard has an area of 315 m2. What is the length of the sides of the backyard? Write your answer as a radical in simplest form and as a decimal rounded to the nearest tenth.

 

 
Practice 6

Annika is standing on a sidewalk at point N and wants to get to the badminton court at point S. If Annika bikes along the sidewalk, she will have to bike 8 m and then 10 m. How far will Annika have to bike if she decides to bike through the field directly from point N to point S? Write your answer as a radical in simplest form and as a decimal rounded to the nearest tenth.

Right Triangle with legs equal to 8 m and 10 m. The right angle is labelled as C. The other angles are labelled N and S.

Simplifying Radicals with Variables

We will now simplify radicals that have variables as part of their radicand.

Example 3

Simplify: \sqrt{{x}^{3}}

Table 2.3.6
Steps Solution

Rewrite the radicand as a product using the largest perfect square factor.

\sqrt{{x}^{3}}

\sqrt{{x}^{2} \bullet x}

Rewrite the radical as the product of two radicals.

\sqrt{{x}^{2}} \bullet \sqrt{x}
Simplify.

x\sqrt{x}
Practice 7

Simplify.

a) \sqrt{{b}^{5}}

 

 

 b) \sqrt{{p}^{9}}

 

 
Example 4

Simplify: \sqrt{72{n}^{7}}

Table 2.3.7
Steps Solution
Rewrite the radicand as a product using the largest perfect square factor. \sqrt{72{n}^{7}}

\sqrt{36{n}^{6} \bullet 2n}
Rewrite the radical as the product of two radicals. \sqrt{36{n}^{6}} \bullet \sqrt{2n}
Simplify. 6{n}^{3}\sqrt{2n}
Practice 8

Simplify.

a) \sqrt{16{x}^{7}}

 

 

 

 b) \sqrt{63{u}^{2}{v}^{5}}

 

 

 

 c) \sqrt{50{a}^{7}{b}^{4}}

 

 

 

Homework

  1. For the following numbers, find the perfect squares that divide evenly into the radicand:

    a) 18

     

    		 b) 75

     

    		 c) 125

     

    		 d) 72

     

    		 e) 98

     

    		 f) 45

     

  2. Convert the following into mixed radicals in simplest form:

    a) \sqrt{12}

     

     

     

     

     

    		 b) \sqrt{72}

     

     

     

     

     

    		 c) -\sqrt{54}

     

     

     

     

     

    		 d) \sqrt{150}

     

     

     

     

     

    		 e) \sqrt{98}

     

     

     

     

     
    f) 5\sqrt{50}

     

     

     

     

     

     

    		 g) \frac{1}{2}\sqrt{51}

     

     

     

     

     

     

    		 h) -6\sqrt{48}

     

     

     

     

     

     

    		 i) 0.2\sqrt{20}

     

     

     

     

     

     

    		 j) \frac{2}{5}\sqrt{150}

     

     

     

     

     

     

  3. Convert the following mixed radicals into entire radicals.

    a) 7\sqrt{3}

     

     

     

     

    		 b) 2\sqrt{5}

     

     

     

     

    		 c) 4\sqrt{5}

     

     

     

     

    		 d) 6\sqrt{3}

     

     

     

     

    		 e) -3\sqrt{7}

     

     

     

     
    f) 0.5\sqrt{100}

     

     

     

     

    		 g) -6\sqrt{13}

     

     

     

     

    		 h) \frac{1}{3}\sqrt{27}

     

     

     

     

    		 i) -0.8\sqrt{25}

     

     

     

     

    		 j) -\frac{3}{2}\sqrt{8}

     

     

     

     

  4. Leila wants to install a square accent designer tile for her new shower. If she can afford to buy one tile that is 625 cm2, what is the side length of the tile?

     

     

     

  5. Anwar wants to have a square mosaic inlaid in his new patio. His budget allows for 2 025 tiles. Each tile is a square with an area of one square inch. How long can a side of the mosaic be?

     

     

     

  6. Yangqi wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. How long can a side of his garden be? Round to one decimal place.

     

     

     

  7. A rope is needed to support a tree that is 12 feet tall. The rope will be anchored to a stake 8 feet from the base of the tree. How much rope is needed? Write your answer as a radical in simplest form and as a decimal rounded to the nearest hundredth.

     

     

     

  8. Starting at her home, a runner runs 3 km west and then 6 km north. What is the shortest distance she can run to return home? Write your answer as a radical in simplest form and as a decimal rounded to the nearest hundredth.

     

     

     

  9. Simplify.

    a) \sqrt{{a}^{2}}

     

     

    		 b) \sqrt{{c}^{7}}

     

     

    		 c) \sqrt{{n}^{4}}

     

     

    		 d) \sqrt{{x}^{3}}

     

     

    		 e) \sqrt{{p}^{9}}

     

     

  10. Simplify.

    a) -7\sqrt{64x^4}

     

     

     

    		 b) -2\sqrt{128n}

     

     

     

    		 c) -5\sqrt{36m}

     

     

     

    		 d) 8\sqrt{112p^2}

     

     

     

  11. Simplify.

    a) \sqrt{45x^2y^2}

     

     

     

     

    		 b) \sqrt{72a^3b^4}

     

     

     

     

    		 c) \sqrt{16x^3y^3}

     

     

     

     
    d) \sqrt{512a^4b^2}

     

     

     

     

     

    		 e) \sqrt{320x^4y^4}

     

     

     

     

     

Answers

1.

a) 9 × 2 b) 25 × 3 c) 25 × 5 d) 36 × 2 e) 49 × 2 f) 9 × 5

2.

a) 2\sqrt{3} b) 6\sqrt{2} c) -3\sqrt{6} d) 5\sqrt{6} e) 7\sqrt{2}
f) 25\sqrt{2} g) cannot be simplified h) -24\sqrt{3} i) 0.4\sqrt{5} j) 2\sqrt{6}

3.

a) \sqrt{147} b) \sqrt{20} c) \sqrt{80} d) \sqrt{108} e) -\sqrt{63}
f) \sqrt{25} g) -\sqrt{468} h) \sqrt{3} i) -\sqrt{16} j) -\sqrt{18}
4. 25 cm 5. 45 in 6. 8.7 ft 7. 4\sqrt{13} ft, 14.42 ft 8. 3\sqrt{5} km, 6.71 km

9.

a) a b) c^3\sqrt{c} c) n2  d) x\sqrt{x} e) p^4\sqrt{p}

10.

a) -56x^2 b) -16\sqrt{2n} c) -30\sqrt{m} d) 32p\sqrt{7}

11.

a) 3xy\sqrt{5} b) 6ab^2\sqrt{2a} c) 4xy\sqrt{xy} d) 16a^2b\sqrt{2} e) 8x^2y^2\sqrt{5}

 

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resources:

Berg, T. (2020). Intermediate algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY-NC-SA 4.0

Marecek, L., & Honeycutt Mathis, A. (2020). Intermediate algebra 2e. OpenStax. https://collection.bccampus.ca/textbooks/key-concepts-of-intermediate-level-math-bccampus-204/, licensed under CC BY 4.0

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

 

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