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

3 <p>If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in the field of vehicle design, finance, etc. Here, we will discuss the square root of 420.</p>

4 <h2>What is the Square Root of 420?</h2>

5 <p>The<a>square</a>root is the inverse<a>of</a>the square of a<a>number</a>. 420 is not a<a>perfect square</a>. The square root of 420 can be expressed in both radical and exponential forms. In the radical form, it is expressed as √420, whereas in<a>exponential form</a>it is (420)^(1/2). √420 ≈ 20.4939, which is an<a>irrational number</a>because it cannot be expressed in the form of p/q, where p and q are<a>integers</a>and q ≠ 0.</p>

6 <h2>Finding the Square Root of 420</h2>

7 <p>The<a>prime factorization</a>method is 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 </ul><ul><li>Long division method</li>

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

11 </ul><h2>Square Root of 420 by Prime Factorization Method</h2>

12 <p>The<a>product</a>of prime<a>factors</a>is the prime factorization of a number. Now let us look at how 420 is broken down into its prime factors:</p>

13 <p><strong>Step 1:</strong>Finding the prime factors of 420 Breaking it down, we get 2 × 2 × 3 × 5 × 7: 2^2 × 3^1 × 5^1 × 7^1</p>

14 <p><strong>Step 2:</strong>Now we found out the prime factors of 420. The second step is to make pairs of those prime factors. Since 420 is not a perfect square, the digits of the number can’t be grouped in pairs completely.</p>

15 <p><strong>Therefore, calculating the<a>square root</a>of 420 using prime factorization gives an approximation.</strong></p>

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18 <h2>Square Root of 420 by Long Division Method</h2>

17 <h2>Square Root of 420 by Long Division Method</h2>

19 <p>The long<a>division</a>method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square numbers for the given number. Let us now learn how to find the square root using the long division method, step by step:</p>

18 <p>The long<a>division</a>method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square numbers for the given number. Let us now learn how to find the square root using the long division method, step by step:</p>

20 <p><strong>Step 1:</strong>To begin with, group the numbers from right to left. In the case of 420, we group it as 20 and 4.</p>

19 <p><strong>Step 1:</strong>To begin with, group the numbers from right to left. In the case of 420, we group it as 20 and 4.</p>

21 <p><strong>Step 2:</strong>Find n whose square is<a>less than</a>or equal to 4. We can say n is 2 because 2 × 2 = 4. The<a>quotient</a>is 2, and after subtracting 4 - 4, the<a>remainder</a>is 0.</p>

20 <p><strong>Step 2:</strong>Find n whose square is<a>less than</a>or equal to 4. We can say n is 2 because 2 × 2 = 4. The<a>quotient</a>is 2, and after subtracting 4 - 4, the<a>remainder</a>is 0.</p>

22 <p><strong>Step 3:</strong>Bring down 20, which becomes the new<a>dividend</a>. Add the old<a>divisor</a>with itself: 2 + 2 = 4, which will be our new divisor.</p>

21 <p><strong>Step 3:</strong>Bring down 20, which becomes the new<a>dividend</a>. Add the old<a>divisor</a>with itself: 2 + 2 = 4, which will be our new divisor.</p>

23 <p><strong>Step 4:</strong>We find 4n such that 4n × n ≤ 20. If n is 0, then 4 × 0 = 0.</p>

22 <p><strong>Step 4:</strong>We find 4n such that 4n × n ≤ 20. If n is 0, then 4 × 0 = 0.</p>

24 <p><strong>Step 5:</strong>Subtract 0 from 20, yielding a remainder of 20.</p>

23 <p><strong>Step 5:</strong>Subtract 0 from 20, yielding a remainder of 20.</p>

25 <p><strong>Step 6:</strong>Add a<a>decimal</a>point to the quotient and bring down two zeros to make the new dividend 2000.</p>

24 <p><strong>Step 6:</strong>Add a<a>decimal</a>point to the quotient and bring down two zeros to make the new dividend 2000.</p>

26 <p><strong>Step 7:</strong>The new divisor is 40n. Find n such that 40n × n is less than or equal to 2000. If n is 4, then 404 × 4 = 1616.</p>

25 <p><strong>Step 7:</strong>The new divisor is 40n. Find n such that 40n × n is less than or equal to 2000. If n is 4, then 404 × 4 = 1616.</p>

27 <p><strong>Step 8:</strong>Subtract 1616 from 2000, leaving a remainder of 384.</p>

26 <p><strong>Step 8:</strong>Subtract 1616 from 2000, leaving a remainder of 384.</p>

28 <p><strong>Step 9:</strong>Repeat the process until the desired decimal places are obtained.</p>

27 <p><strong>Step 9:</strong>Repeat the process until the desired decimal places are obtained.</p>

29 <p><strong>The square root of 420 is approximately 20.49.</strong></p>

28 <p><strong>The square root of 420 is approximately 20.49.</strong></p>

30 <h2>Square Root of 420 by Approximation Method</h2>

29 <h2>Square Root of 420 by Approximation Method</h2>

31 <p>The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Let us learn how to find the square root of 420 using the approximation method.</p>

30 <p>The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Let us learn how to find the square root of 420 using the approximation method.</p>

32 <p><strong>Step 1:</strong>Find the closest perfect squares to √420. The closest perfect squares are 400 (20^2) and 441 (21^2). Thus, √420 falls between 20 and 21.</p>

31 <p><strong>Step 1:</strong>Find the closest perfect squares to √420. The closest perfect squares are 400 (20^2) and 441 (21^2). Thus, √420 falls between 20 and 21.</p>

33 <p><strong>Step 2:</strong>Apply the<a>formula</a>: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (420 - 400) / (441 - 400) = 20 / 41 ≈ 0.49. Add this decimal to the smaller<a>whole number</a>: 20 + 0.49 = 20.49.</p>

32 <p><strong>Step 2:</strong>Apply the<a>formula</a>: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (420 - 400) / (441 - 400) = 20 / 41 ≈ 0.49. Add this decimal to the smaller<a>whole number</a>: 20 + 0.49 = 20.49.</p>

34 <p><strong>Thus, the square root of 420 is approximately 20.49.</strong></p>

33 <p><strong>Thus, the square root of 420 is approximately 20.49.</strong></p>

35 <h2>Common Mistakes and How to Avoid Them in the Square Root of 420</h2>

34 <h2>Common Mistakes and How to Avoid Them in the Square Root of 420</h2>

36 <p>Students do make mistakes while finding the square root, such as forgetting the negative square root, skipping long division steps, etc. Let us look at a few common mistakes students tend to make in detail.</p>

35 <p>Students do make mistakes while finding the square root, such as forgetting the negative square root, skipping long division steps, etc. Let us look at a few common mistakes students tend to make in detail.</p>

36 + <h2>Download Worksheets</h2>

37 <h3>Problem 1</h3>

38 <p>Can you help Max find the area of a square box if its side length is given as √420?</p>

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

40 <p>The area of the square is 420 square units.</p>

41 <h3>Explanation</h3>

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

43 <p>The side length is given as √420.</p>

44 <p>Area of the square = side² = √420 × √420 = 420.</p>

45 <p>Therefore, the area of the square box is 420 square units.</p>

46 <p>Well explained 👍</p>

47 <h3>Problem 2</h3>

48 <p>A square-shaped building measuring 420 square feet is built; if each of the sides is √420, what will be the square feet of half of the building?</p>

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

50 <p>210 square feet.</p>

51 <h3>Explanation</h3>

52 <p>We can just divide the given area by 2 as the building is square-shaped.</p>

53 <p>Dividing 420 by 2 = 210.</p>

54 <p>So half of the building measures 210 square feet.</p>

55 <p>Well explained 👍</p>

56 <h3>Problem 3</h3>

57 <p>Calculate √420 × 5.</p>

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

59 <p>102.47</p>

60 <h3>Explanation</h3>

61 <p>The first step is to find the square root of 420, which is approximately 20.49.</p>

62 <p>The second step is to multiply 20.49 by 5. So, 20.49 × 5 ≈ 102.47.</p>

63 <p>Well explained 👍</p>

64 <h3>Problem 4</h3>

65 <p>What will be the square root of (420 + 4)?</p>

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

67 <p>The square root is 20.6.</p>

68 <h3>Explanation</h3>

69 <p>To find the square root, we need to find the sum of (420 + 4).</p>

70 <p>420 + 4 = 424, and then √424 ≈ 20.6.</p>

71 <p>Therefore, the square root of (420 + 4) is 20.6.</p>

72 <p>Well explained 👍</p>

73 <h3>Problem 5</h3>

74 <p>Find the perimeter of the rectangle if its length ‘l’ is √420 units and the width ‘w’ is 38 units.</p>

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

76 <p>The perimeter of the rectangle is 117.98 units.</p>

77 <h3>Explanation</h3>

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

79 <p>Perimeter = 2 × (√420 + 38) ≈ 2 × (20.49 + 38) = 2 × 58.49 ≈ 117.98 units.</p>

80 <p>Well explained 👍</p>

81 <h2>FAQ on Square Root of 420</h2>

82 <h3>1.What is √420 in its simplest form?</h3>

83 <p>The prime factorization of 420 is 2 × 2 × 3 × 5 × 7, so the simplest form of √420 = √(2 × 2 × 3 × 5 × 7).</p>

84 <h3>2.Mention the factors of 420.</h3>

85 <p>Factors of 420 are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, and 420.</p>

86 <h3>3.Calculate the square of 420.</h3>

87 <p>We get the square of 420 by multiplying the number by itself: 420 × 420 = 176,400.</p>

88 <h3>4.Is 420 a prime number?</h3>

89 <h3>5.420 is divisible by?</h3>

90 <p>420 has many factors; those are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, and 420.</p>

91 <h2>Important Glossaries for the Square Root of 420</h2>

92 <ul><li><strong>Square root:</strong>A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the square root of 16 is √16 = 4.</li>

93 </ul><ul><li><strong>Irrational number:</strong>An irrational number is a number that cannot be expressed as a fraction of two integers (p/q) where q is not zero.</li>

94 </ul><ul><li><strong>Radical form:</strong>A way of expressing numbers using the root symbol, such as √420.</li>

95 </ul><ul><li><strong>Perfect square:</strong>A number that is the square of an integer. For example, 16 is a perfect square because it is 4 squared (4 × 4).</li>

96 </ul><ul><li><strong>Approximation:</strong>The process of finding a value that is close to the exact value, often used with non-perfect squares.</li>

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

98 <p>▶</p>

99 <h2>Jaskaran Singh Saluja</h2>

100 <h3>About the Author</h3>

101 <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>

102 <h3>Fun Fact</h3>

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