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1 - <p>35182 Learners</p>

1 + <p>40251 Learners</p>

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

2 + <p>Last updated on<strong>January 16, 2026</strong></p>

3 <p>The square root of 123 is a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 123. The number 123 has a unique non-negative square root, the principal square root.</p>

4 <h2>What Is the Square Root of 123?</h2>

5 <p>The<a>square</a>root of 123 is ±11.0905365, where 11.0905365 is the positive solution of the<a>equation</a>x2 = 123.</p>

6 <p>Finding the square root is just the inverse of squaring a<a>number</a>and hence, squaring 11.0905365 will result in 123.</p>

7 <p>The square root of 123 is written as √123 in radical form, where the ‘√’ sign is called the “radical” sign. In<a>exponential form</a>, it is written as (123)1/2</p>

8 <h2>Finding the Square Root of 123</h2>

9 <p>We can find the<a>square root</a>of 123 through various methods. They are:</p>

10 <p>i) Prime factorization method</p>

11 <p>ii) Long<a>division</a>method</p>

12 <p>iii) Approximation/Estimation method </p>

13 <h3>Square Root of 123 By Prime Factorization Method</h3>

14 <p>The<a>prime factorization</a>of 123 can be found by dividing 123 by<a>prime numbers</a>and continuing to divide the quotients until they can’t be separated anymore. After factorizing 123, make pairs out of the<a>factors</a>to get the square root. If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the acquired pairs</p>

15 <p>So, Prime factorization of 123 = 41 × 3 </p>

16 <p>But here in the case of 123, no pair of factors are obtained but a single 3 and a single 41 are remaining</p>

17 <p>So, it can be expressed as √123 = √(41 × 3) = √123</p>

18 <p>√123 is the simplest radical form of √123 </p>

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21 <h3>Square Root of 123 By Long Division Method</h3>

20 <h3>Square Root of 123 By Long Division Method</h3>

22 <p>This method is 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, where the dividend is the number we are finding the square root of.</p>

21 <p>This method is 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, where the dividend is the number we are finding the square root of.</p>

23 <p>Follow the steps to calculate the square root of 123:</p>

22 <p>Follow the steps to calculate the square root of 123:</p>

24 <p><strong>Step 1:</strong>Place the number 123 just the same as the image, starting from right to left, and draw a bar above the pair of digits.</p>

23 <p><strong>Step 1:</strong>Place the number 123 just the same as the image, starting from right to left, and draw a bar above the pair of digits.</p>

25 <p><strong> Step 2:</strong> The first number under “bar” is </p>

24 <p><strong> Step 2:</strong> The first number under “bar” is </p>

26 <p>Now, find the greatest number whose square is<a>less than</a>or equal to 1. Here, it is 1, Because 1²=1.</p>

25 <p>Now, find the greatest number whose square is<a>less than</a>or equal to 1. Here, it is 1, Because 1²=1.</p>

27 <p><strong>Step 3: </strong>now divide 1 by 1 (the number we got from Step 2) such that we get 1 as a largest possible number A1=1 is chosen such that when 1 is written beside the new divisor 2, a 2-digit number is formed →21, and multiplying 1 with 21 gives 21, whichis less than 23.</p>

26 <p><strong>Step 3: </strong>now divide 1 by 1 (the number we got from Step 2) such that we get 1 as a largest possible number A1=1 is chosen such that when 1 is written beside the new divisor 2, a 2-digit number is formed →21, and multiplying 1 with 21 gives 21, whichis less than 23.</p>

28 <p> Repeat this process until you reach the remainder of 0. We are left with thethe remainder, 11900 (refer to the picture), after some iterations and keeping the division till here, at this point</p>

27 <p> Repeat this process until you reach the remainder of 0. We are left with thethe remainder, 11900 (refer to the picture), after some iterations and keeping the division till here, at this point</p>

29 <p><strong> Step 4:</strong>The quotient obtained is the square root. In this case, it is 11.090…. </p>

28 <p><strong> Step 4:</strong>The quotient obtained is the square root. In this case, it is 11.090…. </p>

30 <h3>Square Root of 123 By Approximation</h3>

29 <h3>Square Root of 123 By Approximation</h3>

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

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

32 <p><strong>Step 1:</strong>find the square roots of the perfect squares above and below 123</p>

31 <p><strong>Step 1:</strong>find the square roots of the perfect squares above and below 123</p>

33 <p>Below : 121 → square root of 121 = 11 ….(i)</p>

32 <p>Below : 121 → square root of 121 = 11 ….(i)</p>

34 <p>Above : 144 →square root of 144 = 12 …..(ii)</p>

33 <p>Above : 144 →square root of 144 = 12 …..(ii)</p>

35 <p><strong>Step 2:</strong>Dividing 123 with one of 11 or 12 </p>

34 <p><strong>Step 2:</strong>Dividing 123 with one of 11 or 12 </p>

36 <p> If we choose 11 and divide 123 by 11, we are getting 11.18181 ….(iii)</p>

35 <p> If we choose 11 and divide 123 by 11, we are getting 11.18181 ….(iii)</p>

37 <p><strong>Step 3: </strong>find the<a>average</a>of 11 (from Step (i)) and 11.18181 (from Step (iii))</p>

36 <p><strong>Step 3: </strong>find the<a>average</a>of 11 (from Step (i)) and 11.18181 (from Step (iii))</p>

38 <p>(11+11.18181)/2 = 11.0909 </p>

37 <p>(11+11.18181)/2 = 11.0909 </p>

39 <p>Hence, 11.0909 is the approximate square root of 123 </p>

38 <p>Hence, 11.0909 is the approximate square root of 123 </p>

40 <h2>Common Mistakes and How to Avoid Them in the Square Root of 123</h2>

39 <h2>Common Mistakes and How to Avoid Them in the Square Root of 123</h2>

41 <p>When we find the square root of 123, 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 <p>When we find the square root of 123, 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>

41 + <h2>Download Worksheets</h2>

42 <h3>Problem 1</h3>

43 <p>Estimate the value of √123 using an initial guess of 11.08 Solution: using the formula, New Guess=(Initial Guess + (Given Number / Initial Guess))/ 2</p>

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

45 <p>Applying the formula, </p>

46 <p>New guess= (11.08 + (123/11.08))/2 </p>

47 <p>= (11.08+ 11.1010)/2</p>

48 <p>=22.181/2</p>

49 <p> =11.0905</p>

50 <p>Again, repeating the process,</p>

51 <p>New guess= (11.0905 + (123/11.0905))/2 </p>

52 <p>= (11.0905+ 11.09057)/2</p>

53 <p> =22.1855/2</p>

54 <p>=11.09278</p>

55 <p>hence, after a few iterations, the value of √123 is approximately 11.09278</p>

56 <p>Answer: 11.09278 approx </p>

57 <h3>Explanation</h3>

58 <p>Using the formula for New Guess, we found the approximate value of the square root 123 by repeated iterations, where New Guess=(Initial Guess + (Given Number / Initial Guess))/ 2 </p>

59 <p>Well explained 👍</p>

60 <h3>Problem 2</h3>

61 <p>Find the length of a side of a square whose area is 123 cm²</p>

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

63 <p>Given, the area = 123 cm2</p>

64 <p>We know that, (side of a square)2 = area of square</p>

65 <p>Or, (side of a square)2 = 123</p>

66 <p> Or, (side of a square) = √123</p>

67 <p>Or, side of a square = 11.0905.</p>

68 <p>But, the length of a square is a positive quantity only, so, the length of the side is 11.0905 cm.</p>

69 <p>Answer: 11.0905 cm </p>

70 <h3>Explanation</h3>

71 <p>We know that, (side of a square)2 = area of square. Here, we are given with the area of the square, so, we can easily find out its square root because its square root is the measure of the side of the square</p>

72 <p>Well explained 👍</p>

73 <h3>Problem 3</h3>

74 <p>Simplify (√123 + √123) ⤫ √123</p>

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

76 <p> (√123 + √123) ⤫ √123</p>

77 <p>= (11.0905 + 11.0905) ⤫ 11.0905</p>

78 <p>= 22.181 ⤫ 11.0905</p>

79 <p>= 245.998</p>

80 <p>Answer: 245.998 </p>

81 <h3>Explanation</h3>

82 <p>We first solved the part inside the brackets, i.e., √123 + √123, which resulted into 22.181, and then multiply it with √123 which is 11.0905 we get 245.998. </p>

83 <p>Well explained 👍</p>

84 <h3>Problem 4</h3>

85 <p>if x= √123, what is x²-3 ?</p>

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

87 <p>x= √123</p>

88 <p>⇒ x2 = 123</p>

89 <p>⇒ x2-3 = 123-3</p>

90 <p>⇒ x2-3 = 120</p>

91 <p>Answer: 120 </p>

92 <h3>Explanation</h3>

93 <p>We squared the given value of x and then subtracted 3 from it.</p>

94 <p>Well explained 👍</p>

95 <h3>Problem 5</h3>

96 <p>Calculate (√123/3 + √123/10)</p>

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

98 <p>√123/3 + √123/10</p>

99 <p>= 11.09053/ 3 + 11.09053/10</p>

100 <p>= 3.6968 + 1.10905</p>

101 <p>= 4.80585</p>

102 <p>Answer: 4.80585 </p>

103 <h3>Explanation</h3>

104 <p>From the given expression, we first found the value of the square root of 123 then solved by simple divisions and then simple addition. </p>

105 <p>Well explained 👍</p>

106 <h2>FAQs on 123 Square Root</h2>

107 <h3>1.Is 123 a prime number?</h3>

108 <p>No, 123 is not a prime number.</p>

109 <h3>2.Is the square root of 123 a whole number?</h3>

110 <p>No, 11.0905365, the square root of 123, is never a<a>whole number</a></p>

111 <h3>3.Is 123 a perfect square or a non-perfect square?</h3>

112 <p>123 is a non-perfect square, since 123 = (11.0905365)2.</p>

113 <h3>4.Is the square root of 123 a rational or irrational number?</h3>

114 <p>The square root of 123 is ±11.0905365. So, 11.0905365 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.</p>

115 <h3>5.What are the factors of 123?</h3>

116 <p>The principal square root of 123 is ±11.0905365, the positive value, but not -11.0905365.</p>

117 <h3>6.Is the square root of 123 a real number?</h3>

118 <h2>Important Glossaries for Square Root of 123</h2>

119 <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, 2 4 = 16, where 2 is the base, 4 is the exponent </li>

120 </ul><ul><li><strong>Factorization: </strong>Expressing the given expression as a product of its factors Ex: 48=2 ⤬ 2 ⤬ 2 ⤬ 2 ⤬ 3</li>

121 </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>

122 </ul><ul><li><strong>Rational numbers and Irrational numbers:</strong> The numbers that can be expressed as p/q, where p and q are integers and q is not equal to 0 are called Rational numbers. Numbers that cannot be expressed as p/q, where p and q are integers and q is not equal to 0 are called Irrational numbers. </li>

123 </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>

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

125 <p>▶</p>

126 <h2>Jaskaran Singh Saluja</h2>

127 <h3>About the Author</h3>

128 <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>

129 <h3>Fun Fact</h3>

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