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2 <p>Last updated on<strong>August 5, 2025</strong></p>

3 <p>The square root of 5 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y × y, the result is 5. It contains both positive and a negative root, where the positive root is called the principal square root.</p>

4 <h2>What Is the Square Root of 5?</h2>

5 <p>The<a>square</a>root<a>of</a>5 is ±2.2360. The positive value, 2.2360 is the solution of the<a>equation</a>x2 = 5. As defined, the square root is just the inverse of squaring a<a>number</a>, so, squaring 2.2360 will result in 5. The square root of 5 is expressed as √5 in radical form, where the ‘√’ sign is called “radical” sign. In<a>exponential form</a>, it is written as (5)1/2 </p>

6 <h2>Finding the Square Root of 5</h2>

7 <p>We can find the<a>square root</a>of 5 through various methods. They are:</p>

8 <ul><li>Prime factorization method</li>

9 </ul><ul><li>Long<a>division</a>method</li>

10 </ul><ul><li>Approximation/Estimation method</li>

11 </ul><h3>Square Root of 5 By Prime Factorization Method</h3>

12 <p>The<a>prime factorization</a>of 5 involves breaking down a number into its<a>factors</a>. Divide 5 by<a>prime numbers</a>, and continue to divide the quotients until they can’t be separated anymore. After factorizing 5, make pairs out of the factors to get the square root. If there exists numbers which cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs</p>

13 <p>So, Prime factorization of 5 = 5 × 1</p>

14 <p> for 5, no pairs of factors can be obtained, but a single 5 is obtained.</p>

15 <p>So, it can be expressed as √5 = √(5 × 1) = √5</p>

16 <p>√5 is the simplest radical form of √5</p>

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19 <h3>Square Root of 5 by Long Division Method</h3>

18 <h3>Square Root of 5 by Long Division Method</h3>

20 <p>This is a method used for obtaining the square root for non-<a>perfect squares</a>, mainly. It usually involves the division of the<a>dividend</a>by the<a>divisor</a>, getting a<a>quotient</a>and a<a>remainder</a>too sometimes.</p>

19 <p>This is a method used for obtaining the square root for non-<a>perfect squares</a>, mainly. It usually involves the division of the<a>dividend</a>by the<a>divisor</a>, getting a<a>quotient</a>and a<a>remainder</a>too sometimes.</p>

21 <p>Follow the steps to calculate the square root of 5:</p>

20 <p>Follow the steps to calculate the square root of 5:</p>

22 <p><strong>Step 1:</strong>Write the number 5, and draw a bar above the pair of digits from right to left.</p>

21 <p><strong>Step 1:</strong>Write the number 5, and draw a bar above the pair of digits from right to left.</p>

23 <p> <strong>Step 2:</strong>Now, find the greatest number whose square is<a>less than</a>or equal to 5. Here, it is 2, Because 22=4 < 5.</p>

22 <p> <strong>Step 2:</strong>Now, find the greatest number whose square is<a>less than</a>or equal to 5. Here, it is 2, Because 22=4 < 5.</p>

24 <p><strong>Step 3 :</strong>Now divide 5 by 2 (the number we got from Step 2) such that we get 2 as quotient and we get a remainder. Double the divisor 2, we get 4, and then the largest possible number A1=2 is chosen such that when 2 is written beside the new divisor, 4, a 2-digit number is formed →42, and multiplying 2 with 42 gives 84 which is less than 100.</p>

23 <p><strong>Step 3 :</strong>Now divide 5 by 2 (the number we got from Step 2) such that we get 2 as quotient and we get a remainder. Double the divisor 2, we get 4, and then the largest possible number A1=2 is chosen such that when 2 is written beside the new divisor, 4, a 2-digit number is formed →42, and multiplying 2 with 42 gives 84 which is less than 100.</p>

25 <p>Repeat the process until you reach the remainder of 0</p>

24 <p>Repeat the process until you reach the remainder of 0</p>

26 <p>We are left with the remainder, 27100 (refer to the picture), after some iterations and keeping the division till here, at this point </p>

25 <p>We are left with the remainder, 27100 (refer to the picture), after some iterations and keeping the division till here, at this point </p>

27 <p> <strong>Step 4 :</strong>The quotient obtained is the square root. In this case, it is 2.23….</p>

26 <p> <strong>Step 4 :</strong>The quotient obtained is the square root. In this case, it is 2.23….</p>

28 <h3>Square Root of 5 by Approximation Method</h3>

27 <h3>Square Root of 5 by Approximation Method</h3>

29 <p>Approximation or<a>estimation</a>of square root is not the exact square root, but it is an estimate.Here, through this method, an approximate value of square root is found by guessing.</p>

28 <p>Approximation or<a>estimation</a>of square root is not the exact square root, but it is an estimate.Here, through this method, an approximate value of square root is found by guessing.</p>

30 <p>Follow the steps below:</p>

29 <p>Follow the steps below:</p>

31 <p><strong>Step 1 :</strong>Identify the square roots of the perfect squares above and below 5</p>

30 <p><strong>Step 1 :</strong>Identify the square roots of the perfect squares above and below 5</p>

32 <p>Below : 4→ square root of 4 = 2 ……..(<a>i</a>)</p>

31 <p>Below : 4→ square root of 4 = 2 ……..(<a>i</a>)</p>

33 <p>Above : 9 →square root of 9 = 3 ……..(ii)</p>

32 <p>Above : 9 →square root of 9 = 3 ……..(ii)</p>

34 <p><strong>Step 2 :</strong>Divide 5 with one of 2 or 3</p>

33 <p><strong>Step 2 :</strong>Divide 5 with one of 2 or 3</p>

35 <p>If we choose 2, and divide 5 by 2, we get 2.5 …….(iii)</p>

34 <p>If we choose 2, and divide 5 by 2, we get 2.5 …….(iii)</p>

36 <p> <strong>Step 3:</strong>Find the<a>average</a>of 2 (from (i)) and 2.5 (from (iii))</p>

35 <p> <strong>Step 3:</strong>Find the<a>average</a>of 2 (from (i)) and 2.5 (from (iii))</p>

37 <p>(2+2.5)/2 = 2.25</p>

36 <p>(2+2.5)/2 = 2.25</p>

38 <p> Hence, 2.25 is the approximate square root of 5 </p>

37 <p> Hence, 2.25 is the approximate square root of 5 </p>

39 <h2>Common Mistakes and How to Avoid Them in the Square Root of 5</h2>

38 <h2>Common Mistakes and How to Avoid Them in the Square Root of 5</h2>

40 <p>When we find the square root of 5, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.</p>

39 <p>When we find the square root of 5, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.</p>

40 + <h2>Download Worksheets</h2>

41 <h3>Problem 1</h3>

42 <p>Simplify √5 + 5√5 + 10√5 ?</p>

43 <p>Okay, lets begin</p>

44 <p> √5 + 5√5+10√5</p>

45 <p>= √5(1+5+10)</p>

46 <p>= 16√5</p>

47 <p>Answer : 16√5 </p>

48 <h3>Explanation</h3>

49 <p>The simplest radical form of √5 is √5, so, it is taken common outside and calculated simply. </p>

50 <p>Well explained 👍</p>

51 <h3>Problem 2</h3>

52 <p>What is √5 multiplied by 5√5?</p>

53 <p>Okay, lets begin</p>

54 <p>√5 ⤬ 5√5</p>

55 <p>= 5⤬5</p>

56 <p>= 25</p>

57 <p>Answer: 25 </p>

58 <h3>Explanation</h3>

59 <p> √5 multiplying with itself gives 5, and then again multiplied by 5 </p>

60 <p>Well explained 👍</p>

61 <h3>Problem 3</h3>

62 <p>Find the value of 1/√5?</p>

63 <p>Okay, lets begin</p>

64 <p>1/√ 5</p>

65 <p>= 1/ 2.23</p>

66 <p>=0.4291</p>

67 <p>Answer: 0.4291 </p>

68 <h3>Explanation</h3>

69 <p>we divide 1 by the value of √5 </p>

70 <p>Well explained 👍</p>

71 <h3>Problem 4</h3>

72 <p>If y=√5, find y²</p>

73 <p>Okay, lets begin</p>

74 <p>firstly, y=√5</p>

75 <p>= 2.2360</p>

76 <p> Now, squaring y, we get, </p>

77 <p>y2= (2.2360)2</p>

78 <p>=5</p>

79 <p>or, y2=5</p>

80 <p>Answer : 5 </p>

81 <h3>Explanation</h3>

82 <p> squaring “y” which is same as squaring the value of √5 resulted to 5 </p>

83 <p>Well explained 👍</p>

84 <h3>Problem 5</h3>

85 <p>Approximate √5 up to three decimal places.</p>

86 <p>Okay, lets begin</p>

87 <p>√5 ≅ 2.236 (up to three decimal places)</p>

88 <p>Answer : 2.236 </p>

89 <h3>Explanation</h3>

90 <p> finding the square root of 5 up till three decimal places. </p>

91 <p>Well explained 👍</p>

92 <h2>FAQs on 5 Square Root</h2>

93 <h3>1.Is √5 a real number?</h3>

94 <h3>2.Is √5 a whole number ?</h3>

95 <h3>3.Is 5 a perfect square or non-perfect square?</h3>

96 <p> 5 is a non-perfect square, since 5 =(2.2360) 2. </p>

97 <h3>4.Is the square root of 5 a rational or irrational number?</h3>

98 <p>The square root of 5 is ±2.2360. So, 2.2360 is an<a>irrational number</a>, since it cannot be obtained by dividing two<a>integers</a>and cannot be written in the form p/q, where p and q are integers. </p>

99 <h3>5. How to solve √10?</h3>

100 <p>√10 can be solved by methods to find square roots:</p>

101 <p>Long Division Method, Prime Factorization, Approximation method. The value of √10 is 3.162277 </p>

102 <h2>Important Glossaries for Square Root of 5</h2>

103 <p><strong>Exponential form:</strong> An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent. Ex: 2 × 2 × 2 × 2 = 16 Or, 2 4 = 16, where 2 is the base, 4 is the exponent </p>

104 <p><strong>Prime Factorization: </strong> Expressing the given expression as a product of its factors. Ex: 48=2 × 2 × 2 × 2 × 3</p>

105 <p><strong>Prime Numbers:</strong>Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....</p>

106 <p><strong>Rational numbers and Irrational numbers:</strong>The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers. Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. </p>

107 <p><strong>Perfect and non-perfect square numbers:</strong>Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :3, 8, 24 </p>

108 <p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>

109 <p>▶</p>

110 <h2>Jaskaran Singh Saluja</h2>

111 <h3>About the Author</h3>

112 <p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>

113 <h3>Fun Fact</h3>

114 <p>: He loves to play the quiz with kids through algebra to make kids love it.</p>