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1 - <p>292 Learners</p>

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

3 <p>If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The square root is used in fields such as vehicle design and finance. Here, we will discuss the square root of 401.</p>

4 <h2>What is the Square Root of 401?</h2>

5 <p>The<a>square</a>root is the inverse of the square of a<a>number</a>. 401 is not a<a>perfect square</a>. The square root of 401 is expressed in both radical and<a>exponential form</a>. In radical form, it is expressed as √401, whereas in exponential form, it is (401)^(1/2). √401 ≈ 20.02498, which is an<a>irrational number</a>because it cannot be expressed in the form p/q, where p and q are<a>integers</a>and q ≠ 0.</p>

6 <h2>Finding the Square Root of 401</h2>

7 <p>The<a>prime factorization</a>method can be used for perfect square numbers. However, for non-perfect square numbers, the<a>long division</a>method and approximation method are used. Let us now learn the following methods:</p>

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

9 <li>Long division method</li>

10 <li>Approximation method</li>

11 </ul><h2>Square Root of 401 by Prime Factorization Method</h2>

12 <p>Prime factorization involves expressing a number as the<a>product</a>of its prime<a>factors</a>. Since 401 is a<a>prime number</a>, it cannot be broken down into other prime factors. Therefore, calculating the<a>square root</a>of 401 using prime factorization is not possible.</p>

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15 <h2>Square Root of 401 by Long Division Method</h2>

14 <h2>Square Root of 401 by Long Division Method</h2>

16 <p>The long<a>division</a>method is particularly useful for non-perfect square numbers. Here's how to find the square root of 401 using the long division method, step by step:</p>

15 <p>The long<a>division</a>method is particularly useful for non-perfect square numbers. Here's how to find the square root of 401 using the long division method, step by step:</p>

17 <p><strong>Step 1:</strong>Group the digits of 401 from right to left. In the case of 401, we have one group of three digits: 401.</p>

16 <p><strong>Step 1:</strong>Group the digits of 401 from right to left. In the case of 401, we have one group of three digits: 401.</p>

18 <p><strong>Step 2:</strong>Find the largest number whose square is<a>less than</a>or equal to 401. The closest perfect square is 400, and its square root is 20.</p>

17 <p><strong>Step 2:</strong>Find the largest number whose square is<a>less than</a>or equal to 401. The closest perfect square is 400, and its square root is 20.</p>

19 <p><strong>Step 3:</strong>Subtract the square of 20 from 401, which gives a<a>remainder</a>of 1. The<a>quotient</a>is 20.</p>

18 <p><strong>Step 3:</strong>Subtract the square of 20 from 401, which gives a<a>remainder</a>of 1. The<a>quotient</a>is 20.</p>

20 <p><strong>Step 4:</strong>Bring down a pair of zeroes to make the<a>dividend</a>100.</p>

19 <p><strong>Step 4:</strong>Bring down a pair of zeroes to make the<a>dividend</a>100.</p>

21 <p><strong>Step 5:</strong>Double the quotient (20) which gives us a new<a>divisor</a>of 40.</p>

20 <p><strong>Step 5:</strong>Double the quotient (20) which gives us a new<a>divisor</a>of 40.</p>

22 <p><strong>Step 6:</strong>Find a digit n such that 40n × n is less than or equal to 100. The suitable n is 2, as 402 × 2 = 80.</p>

21 <p><strong>Step 6:</strong>Find a digit n such that 40n × n is less than or equal to 100. The suitable n is 2, as 402 × 2 = 80.</p>

23 <p><strong>Step 7:</strong>Subtract 80 from 100, leaving a remainder of 20.</p>

22 <p><strong>Step 7:</strong>Subtract 80 from 100, leaving a remainder of 20.</p>

24 <p><strong>Step 8:</strong>Continue the process by bringing down more pairs of zeroes and repeating the steps above to get more<a>decimal</a>places in the quotient.</p>

23 <p><strong>Step 8:</strong>Continue the process by bringing down more pairs of zeroes and repeating the steps above to get more<a>decimal</a>places in the quotient.</p>

25 <p>The square root of 401 is approximately 20.02498.</p>

24 <p>The square root of 401 is approximately 20.02498.</p>

26 <h2>Square Root of 401 by Approximation Method</h2>

25 <h2>Square Root of 401 by Approximation Method</h2>

27 <p>The approximation method is another way to find square roots. It is an easy method for estimating the square root of a number. Here's how to approximate the square root of 401:</p>

26 <p>The approximation method is another way to find square roots. It is an easy method for estimating the square root of a number. Here's how to approximate the square root of 401:</p>

28 <p><strong>Step 1:</strong>Identify the closest perfect squares around 401.</p>

27 <p><strong>Step 1:</strong>Identify the closest perfect squares around 401.</p>

29 <p>The closest are 400 (20²) and 441 (21²).</p>

28 <p>The closest are 400 (20²) and 441 (21²).</p>

30 <p>√401 falls between 20 and 21.</p>

29 <p>√401 falls between 20 and 21.</p>

31 <p><strong>Step 2:</strong>Use the<a>formula</a>:</p>

30 <p><strong>Step 2:</strong>Use the<a>formula</a>:</p>

32 <p>(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)</p>

31 <p>(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)</p>

33 <p>For 401, this is (401 - 400) / (441 - 400) = 1 / 41 ≈ 0.02439.</p>

32 <p>For 401, this is (401 - 400) / (441 - 400) = 1 / 41 ≈ 0.02439.</p>

34 <p><strong>Step 3:</strong>Add this result to the smaller root: 20 + 0.02439 ≈ 20.02439.</p>

33 <p><strong>Step 3:</strong>Add this result to the smaller root: 20 + 0.02439 ≈ 20.02439.</p>

35 <p>Therefore, the square root of 401 is approximately 20.02439.</p>

34 <p>Therefore, the square root of 401 is approximately 20.02439.</p>

36 <h2>Common Mistakes and How to Avoid Them in the Square Root of 401</h2>

35 <h2>Common Mistakes and How to Avoid Them in the Square Root of 401</h2>

37 <p>Students often make mistakes while finding square roots, such as forgetting about the negative square root or skipping long division steps. Let's examine a few common mistakes in more detail.</p>

36 <p>Students often make mistakes while finding square roots, such as forgetting about the negative square root or skipping long division steps. Let's examine a few common mistakes in more detail.</p>

37 + <h2>Download Worksheets</h2>

38 <h3>Problem 1</h3>

39 <p>Can you help Max find the area of a square box if its side length is given as √401?</p>

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

41 <p>The area of the square is approximately 401 square units.</p>

42 <h3>Explanation</h3>

43 <p>The area of the square = side².</p>

44 <p>The side length is given as √401.</p>

45 <p>Area of the square = (√401)² = 401.</p>

46 <p>Therefore, the area of the square box is approximately 401 square units.</p>

47 <p>Well explained 👍</p>

48 <h3>Problem 2</h3>

49 <p>A square-shaped garden measures 401 square feet. If each side is √401 feet long, what will be the area of half of the garden?</p>

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

51 <p>200.5 square feet</p>

52 <h3>Explanation</h3>

53 <p>Since the garden is square-shaped, divide the total area by 2. 401 ÷ 2 = 200.5.</p>

54 <p>So, half of the garden measures 200.5 square feet.</p>

55 <p>Well explained 👍</p>

56 <h3>Problem 3</h3>

57 <p>Calculate √401 × 5.</p>

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

59 <p>Approximately 100.12</p>

60 <h3>Explanation</h3>

61 <p>First, find the square root of 401, which is approximately 20.02498.</p>

62 <p>Then multiply it by 5. 20.02498 × 5 ≈ 100.12.</p>

63 <p>Well explained 👍</p>

64 <h3>Problem 4</h3>

65 <p>What will be the square root of (401 + 4)?</p>

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

67 <p>The square root is approximately 20.4206.</p>

68 <h3>Explanation</h3>

69 <p>To find the square root, first calculate the sum: 401 + 4 = 405.</p>

70 <p>Then find the square root: √405 ≈ 20.1246.</p>

71 <p>Therefore, the square root of (401 + 4) is approximately 20.1246.</p>

72 <p>Well explained 👍</p>

73 <h3>Problem 5</h3>

74 <p>Find the perimeter of a rectangle if its length ‘l’ is √401 units and the width ‘w’ is 40 units.</p>

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

76 <p>The perimeter of the rectangle is approximately 120.05 units.</p>

77 <h3>Explanation</h3>

78 <p>Perimeter of the rectangle = 2 × (length + width)</p>

79 <p>Perimeter = 2 × (√401 + 40) ≈ 2 × (20.02498 + 40) ≈ 2 × 60.02498 ≈ 120.05 units.</p>

80 <p>Well explained 👍</p>

81 <h2>FAQ on Square Root of 401</h2>

82 <h3>1.What is √401 in its simplest form?</h3>

83 <p>Since 401 is a prime number, the simplest form of √401 is simply √401.</p>

84 <h3>2.Is 401 a prime number?</h3>

85 <p>Yes, 401 is a prime number, as it has only two factors: 1 and 401.</p>

86 <h3>3.Calculate the square of 401.</h3>

87 <p>To find the square of 401, multiply the number by itself: 401 × 401 = 160,801.</p>

88 <h3>4.How can I approximate the square root of a non-perfect square?</h3>

89 <p>To approximate the square root, identify the closest perfect squares and use them to estimate the root of the non-perfect square.</p>

90 <h3>5.Why is √401 considered an irrational number?</h3>

91 <p>√401 is irrational because it cannot be expressed as a<a>fraction</a>of two integers.</p>

92 <h2>Important Glossaries for the Square Root of 401</h2>

93 <ul><li><strong>Square root:</strong>The square root of a number is a value that, when multiplied by itself, gives the original number. For example, 4² = 16, so √16 = 4. </li>

94 <li><strong>Irrational number:</strong>An irrational number cannot be written as a simple fraction, as it has a non-repeating and non-terminating decimal expansion. </li>

95 <li><strong>Prime number:</strong>A prime number has only two factors-1 and itself. </li>

96 <li><strong>Approximation:</strong>This refers to finding a value that is close to but not exactly equal to the actual value. </li>

97 <li><strong>Long division method:</strong>A technique used to divide large numbers and can be adapted to find square roots of non-perfect squares.</li>

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

99 <p>▶</p>

100 <h2>Jaskaran Singh Saluja</h2>

101 <h3>About the Author</h3>

102 <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>

103 <h3>Fun Fact</h3>

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