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1 - <p>1151 Learners</p>

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

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

5 <p>The<a>square</a>root of 3 is ±1.73205080757. The positive value, 1.73205080757 is the solution of the<a>equation</a>x2 = 3. As defined, the square root is just the inverse of squaring a<a>number</a>, so, squaring 1.73205080757 will result in 3. The square root of 3 is expressed as √3 in radical form, where the ‘√’ sign is called “radical” sign. In<a>exponential form</a>, it is written as (3)1/2 </p>

6 <h2>Finding the Square Root of 3</h2>

7 <p>We can find the<a>square root</a>of 3 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 3 By Prime Factorization Method</h3>

12 <p>The<a>prime factorization</a>of 3 involves breaking down a number into its<a>factors</a>. Divide 3 by<a>prime numbers</a>, and continue to divide the quotients until they can’t be separated anymore. After factorizing 3, 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 3 = 3 × 1 </p>

14 <p>for 3, no pairs of factors can be obtained, only a single 3 is obtained.</p>

15 <p>So, it can be expressed as √3 = √(3 × 1) = √3</p>

16 <p>√3 is the simplest radical form of √3. </p>

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19 <h3>Square Root of 3 by Long Division Method</h3>

18 <h3>Square Root of 3 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 3:</p>

20 <p>Follow the steps to calculate the square root of 3:</p>

22 <p><strong>Step 1:</strong>Write the number 3, and draw a bar above the pair of digits from right to left.</p>

21 <p><strong>Step 1:</strong>Write the number 3, 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 3. Here, it is 1, Because 12=1< 3.</p>

22 <p> <strong>Step 2:</strong>Now, find the greatest number whose square is<a>less than</a>or equal to 3. Here, it is 1, Because 12=1< 3.</p>

24 <p><strong>Step 3 :</strong>Now divide 3 by 1 (the number we got from Step 2) such that we get 1 as quotient and we get a remainder. Double the divisor 1, we get 2, and then the largest possible number A1=7 is chosen such that when 7 is written beside the new divisor, 2, a 2-digit number is formed →27, and multiplying 7 with 27 gives 189 which is less than 200.</p>

23 <p><strong>Step 3 :</strong>Now divide 3 by 1 (the number we got from Step 2) such that we get 1 as quotient and we get a remainder. Double the divisor 1, we get 2, and then the largest possible number A1=7 is chosen such that when 7 is written beside the new divisor, 2, a 2-digit number is formed →27, and multiplying 7 with 27 gives 189 which is less than 200.</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, 176 (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, 176 (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 1.732….</p>

26 <p> <strong>Step 4 :</strong>The quotient obtained is the square root. In this case, it is 1.732….</p>

28 <h3>Square Root of 3 by Approximation Method</h3>

27 <h3>Square Root of 3 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 3</p>

30 <p><strong>Step 1 :</strong>Identify the square roots of the perfect squares above and below 3</p>

32 <p>Below : 1→ square root of 1 = 1 ……..(i)</p>

31 <p>Below : 1→ square root of 1 = 1 ……..(i)</p>

33 <p>Above : 4 →square root of 4 = 2 ……..(ii)</p>

32 <p>Above : 4 →square root of 4 = 2 ……..(ii)</p>

34 <p><strong>Step 2 :</strong>Divide 3 with one of 1 or 2</p>

33 <p><strong>Step 2 :</strong>Divide 3 with one of 1 or 2</p>

35 <p> If we choose 2, and divide 3 by 2, we get 1.5 …….(iii)</p>

34 <p> If we choose 2, and divide 3 by 2, we get 1.5 …….(iii)</p>

36 <p> <strong>Step 3:</strong>Find the<a>average</a>of 2 (from (ii)) and 1.5 (from (iii))</p>

35 <p> <strong>Step 3:</strong>Find the<a>average</a>of 2 (from (ii)) and 1.5 (from (iii))</p>

37 <p>(2+1.5)/2 = 1.75</p>

36 <p>(2+1.5)/2 = 1.75</p>

38 <p> Hence, 1.75 is the approximate square root of 3 </p>

37 <p> Hence, 1.75 is the approximate square root of 3 </p>

39 <h2>Common Mistakes and How to Avoid Them in the Square Root of 3</h2>

38 <h2>Common Mistakes and How to Avoid Them in the Square Root of 3</h2>

40 <p>When we find the square root of 3, 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 3, 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 √3 + 5√3 ?</p>

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

44 <p>√3 + 5√3</p>

45 <p>= √3(1+5)</p>

46 <p>= 6√3</p>

47 <p>Answer : 6√3</p>

48 <h3>Explanation</h3>

49 <p>the simplest radical form of √3 is √3, so, we applied that and solved. </p>

50 <p>Well explained 👍</p>

51 <h3>Problem 2</h3>

52 <p>What is √3 multiplied by 2√3 and then divided by (√3)²?</p>

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

54 <p> (√3 ⤬ 2√3)/(√3)2</p>

55 <p>= (3⤬2)/3</p>

56 <p>= 2</p>

57 <p>Answer: 2 </p>

58 <h3>Explanation</h3>

59 <p>√3 multiplying with itself gives 3, and then again multiplied by 2 in the first step and then again divided by (√3)2=3 </p>

60 <p>Well explained 👍</p>

61 <h3>Problem 3</h3>

62 <p>Find the value of 1/√3?</p>

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

64 <p>1/√3</p>

65 <p>= 1/ 1.73</p>

66 <p>=0.578</p>

67 <p>Answer: 0.578 </p>

68 <h3>Explanation</h3>

69 <p>we divide 1 by the value of √3 </p>

70 <p>Well explained 👍</p>

71 <h3>Problem 4</h3>

72 <p>If y=√3, find y², y^3, y^4</p>

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

74 <p>firstly, y=√3 </p>

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

76 <p>y2= (√3)2=3</p>

77 <p>or, y2=3</p>

78 <p>Similarly, y3=(√3)3=3√3</p>

79 <p>Similarly, y4=(√3)4=((√3)2)2=9</p>

80 <p>Answer : 3, 3√3, 9 </p>

81 <h3>Explanation</h3>

82 <p> squaring “y” which is same as squaring the value of √3 resulted to 3 and hence applied this fact to each problem here. </p>

83 <p>Well explained 👍</p>

84 <h3>Problem 5</h3>

85 <p>Find √3 / √3</p>

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

87 <p>√3/√3</p>

88 <p>= √(3/3)</p>

89 <p>= √1</p>

90 <p>= 1</p>

91 <p>Answer : 1 </p>

92 <h3>Explanation</h3>

93 <p>since the numerator and denominator is same, the answer is 1 </p>

94 <p>Well explained 👍</p>

95 <h2>FAQs on 3 Square Root</h2>

96 <h3>1.What is the value of √3?</h3>

97 <p>The value of √3 is 1.73205080757.</p>

98 <h3>2.What is the value of 3√2?</h3>

99 <p> The value of 3√2 = 3 ╳ √2= 3╳ 1.4142135627 =4.242 </p>

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

101 <p>3 is a non-perfect square, since 3 =(1.73205080757)2. </p>

102 <h3>4.Is the square root of 3 a rational or irrational number?</h3>

103 <p>The square root of 3 is ±1.73205080757. So, 1.73205080757 is an irrational number, 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.Is 3 a perfect cube?</h3>

105 <h2>Important Glossaries for Square Root of 3</h2>

106 <ul><li><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, 24 = 16, where 2 is the base, 4 is the exponent </li>

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

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

109 </ul><ul><li><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. </li>

110 </ul><ul><li><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</li>

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

112 <p>▶</p>

113 <h2>Jaskaran Singh Saluja</h2>

114 <h3>About the Author</h3>

115 <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>

116 <h3>Fun Fact</h3>

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