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

3 <p>The square root of 52 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y × y, the result is 52. 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 52?</h2>

5 <p>The<a>square</a>root<a>of</a>52 is ±7.21110255093. The positive value,7.21110255093 is the solution of the<a>equation</a>x2 = 52. As defined, the square root is just the inverse of squaring a<a>number</a>, so, squaring 7.21110255093 will result in 52. The square root of 52 is expressed as √52 in radical form, where the ‘√’ sign is called “radical” sign. In<a>exponential form</a>, it is written as (52)1/2 </p>

6 <h2>Finding the Square Root of 52</h2>

7 <p>We can find the<a>square root</a>of 52 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 52 By Prime Factorization Method</h3>

12 <p>The<a>prime factorization</a>of 52 involves breaking down a number into its<a>factors</a>. Divide 52 by<a>prime numbers</a>, and continue to divide the quotients until they can’t be separated anymore. After factoring 52, 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 52 = 13 × 2 ×2 </p>

14 <p>for 52, one pairs of factors 2 obtained, but a single 13 is also obtained.</p>

15 <p>So, it can be expressed as √52 = √(2 × 2 × 13) = 2√13</p>

16 <p>2√13 is the simplest radical form of √52</p>

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19 <h3>Square Root of 52 by Long Division Method</h3>

18 <h3>Square Root of 52 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 52:</p>

20 <p>Follow the steps to calculate the square root of 52:</p>

22 <p><strong>Step 1 :</strong>Write the number 52, and draw a bar above the pair of digits from right to left.</p>

21 <p><strong>Step 1 :</strong>Write the number 52, 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 52. Here, it is 7, Because 72=49 < 52</p>

22 <p> <strong>Step 2 :</strong>Now, find the greatest number whose square is<a>less than</a>or equal to 52. Here, it is 7, Because 72=49 < 52</p>

24 <p><strong>Step 3 :</strong>Now divide 52 by 7 (the number we got from Step 2) such that we get 7 as quotient, and we get a remainder. Double the divisor 7, we get 14 and then the largest possible number A1=2 is chosen such that when 2 is written beside the new divisor, 14, a 3-digit number is formed →142 and multiplying 2 with 142 gives 284 which is less than 300.</p>

23 <p><strong>Step 3 :</strong>Now divide 52 by 7 (the number we got from Step 2) such that we get 7 as quotient, and we get a remainder. Double the divisor 7, we get 14 and then the largest possible number A1=2 is chosen such that when 2 is written beside the new divisor, 14, a 3-digit number is formed →142 and multiplying 2 with 142 gives 284 which is less than 300.</p>

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

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

26 <p>We are left with the remainder, 1479 (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, 1479 (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 7.211…</p>

26 <p> <strong>Step 4 :</strong>The quotient obtained is the square root. In this case, it is 7.211…</p>

28 <h3>Square Root of 52 by Approximation Method</h3>

27 <h3>Square Root of 52 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 52</p>

30 <p><strong>Step 1 :</strong>Identify the square roots of the perfect squares above and below 52</p>

32 <p>Below : 49→ square root of 49 = 7 ……..(<a>i</a>)</p>

31 <p>Below : 49→ square root of 49 = 7 ……..(<a>i</a>)</p>

33 <p>Above : 64 →square root of 64= 8 ……..(ii)</p>

32 <p>Above : 64 →square root of 64= 8 ……..(ii)</p>

34 <p><strong>Step 2 :</strong>Divide 52 with one of 7 or 8 </p>

33 <p><strong>Step 2 :</strong>Divide 52 with one of 7 or 8 </p>

35 <p>If we choose 7, and divide 52 by 7, we get 7.428 …….(iii)</p>

34 <p>If we choose 7, and divide 52 by 7, we get 7.428 …….(iii)</p>

36 <p> <strong>Step 3:</strong>Find the<a>average</a>of 7 (from (i)) and 7.428 (from (iii))</p>

35 <p> <strong>Step 3:</strong>Find the<a>average</a>of 7 (from (i)) and 7.428 (from (iii))</p>

37 <p>(7+7.428)/2 = 7.2142</p>

36 <p>(7+7.428)/2 = 7.2142</p>

38 <p> Hence, 7.2142 is the approximate square root of 52</p>

37 <p> Hence, 7.2142 is the approximate square root of 52</p>

39 <h2>Common Mistakes and How to Avoid Them in the Square Root of 52</h2>

38 <h2>Common Mistakes and How to Avoid Them in the Square Root of 52</h2>

40 <p>When we find the square root of 52, 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 52, 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√52 + 13√52 ?</p>

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

44 <p> 5√52+13√52</p>

45 <p>= √52(5+13)</p>

46 <p>= 7.211 ⤬ (5+13)</p>

47 <p>=18 ⤬ 7.211</p>

48 <p>= 129.7998 </p>

49 <p>Answer : 129.7998 </p>

50 <h3>Explanation</h3>

51 <p>Taking out the common part √52, adding the values inside the bracket. √52= 7.2111, so multiplying the square root value with the sum </p>

52 <p>Well explained 👍</p>

53 <h3>Problem 2</h3>

54 <p>√52 with 13</p>

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

56 <p>√52 ⤬ 13</p>

57 <p>= 2√13⤬ 13</p>

58 <p>=26√13</p>

59 <p>Answer : 26√13 </p>

60 <h3>Explanation</h3>

61 <p>multiplying the simplest radical form of √52 with 13. </p>

62 <p>Well explained 👍</p>

63 <h3>Problem 3</h3>

64 <p>Compare √52 and √48</p>

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

66 <p> √52 ≅ 7.2111,</p>

67 <p>√48 ≅ 6.928</p>

68 <p>So, √52 is greater than √48</p>

69 <p>Answer: √52 > √48 </p>

70 <h3>Explanation</h3>

71 <p> finding out the approximate values of √52 and √48 and comparing them</p>

72 <p>Well explained 👍</p>

73 <h3>Problem 4</h3>

74 <p>If y=√52, find y²</p>

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

76 <p> firstly, y=√52= 7.21110255093</p>

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

78 <p>y2= (7.21110255093)2=52</p>

79 <p>or, y2=52</p>

80 <p>Answer : 52 </p>

81 <h3>Explanation</h3>

82 <p>squaring “y” which is same as squaring the value of √52 resulted to 52 </p>

83 <p>Well explained 👍</p>

84 <h3>Problem 5</h3>

85 <p>Find √52 / √48</p>

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

87 <p>√52/√48</p>

88 <p>= √(52/49)</p>

89 <p>= 7.2111 / 6.928</p>

90 <p>= 1.04086316</p>

91 <p>Answer : 1.04086316 </p>

92 <h3>Explanation</h3>

93 <p> dividing the square root value of 52 with that of square root value of 48 </p>

94 <p>Well explained 👍</p>

95 <h2>FAQs on Square Root of 52</h2>

96 <h3>1.How to solve √50?</h3>

97 <p>√50 can be solved through methods like Long Division Method, Prime Factorization method, Approximation method </p>

98 <h3>2.What is 52 divisible by ?</h3>

99 <p>Factors of 52 are: 1,2,4,13,26, and 52 </p>

100 <h3>3.Is 52 a perfect square or non-perfect square?</h3>

101 <p>52 is a non-perfect square, since 52 =(7.21110255093) 2. </p>

102 <h3>4.Is the square root of 52 a rational or irrational number?</h3>

103 <p>The square root of 52 is ±7.21110255093. So, 7.21110255093 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>

104 <h3>5. 52 falls between which two perfect squares?</h3>

105 <p>: 52 falls between perfect squares → 49 and 64 </p>

106 <h3>6. What is the closest perfect square to 52?</h3>

107 <p>49 is the closest perfect square to 52 </p>

108 <h2>Important Glossaries for Square Root of 52</h2>

109 <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>

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

111 <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>

112 <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>

113 <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>

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

115 <p>▶</p>

116 <h2>Jaskaran Singh Saluja</h2>

117 <h3>About the Author</h3>

118 <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>

119 <h3>Fun Fact</h3>

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