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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 various fields such as vehicle design and finance. Here, we will discuss the square root of -164.</p>

4 <h2>What is the Square Root of -164?</h2>

5 <p>The<a>square</a>root is the inverse of the square of a<a>number</a>. Since -164 is a<a>negative number</a>, its square root involves<a>imaginary numbers</a>. The square root of -164 is expressed in both radical and<a>exponential form</a>. In the radical form, it is expressed as √(-164), whereas (-164)^(1/2) in the exponential form. The square root of a negative number is an imaginary number, which can be expressed as i√164, where i is the imaginary unit with the property that i² = -1.</p>

6 <h2>Finding the Square Root of -164</h2>

7 <p>To find the<a>square root</a>of a negative number like -164, we need to recognize that it involves the imaginary unit i. The<a>prime factorization</a>and<a>long division</a>methods do not apply directly to negative numbers since they yield<a>real numbers</a>. However, we can find the square root of the positive counterpart and then multiply it by i. Let's explore the steps:</p>

8 <p>1. Find the square root of 164 using the usual methods.</p>

9 <p>2. Multiply the result by i to account for the negative sign.</p>

10 <h2>Square Root of 164 by Prime Factorization Method</h2>

11 <p>The<a>product</a>of prime<a>factors</a>is the prime factorization of a number. Here’s how 164 is broken down into its prime factors:</p>

12 <p><strong>Step 1:</strong>Finding the prime factors of 164 Breaking it down, we get 2 x 2 x 41: 2² x 41</p>

13 <p><strong>Step 2:</strong>Pair the prime factors. Since 164 is not a<a>perfect square</a>, the digits of the number can’t be grouped into pairs for complete pairs.</p>

14 <p>Thus, the square root of 164 is expressed as 2√41, and for -164 as i(2√41).</p>

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17 <h2>Square Root of 164 by Long Division Method</h2>

16 <h2>Square Root of 164 by Long Division Method</h2>

18 <p>The long<a>division</a>method is typically used for finding the square roots of positive non-perfect square numbers. Here’s how to find the square root of 164 using the long division method, then apply the result to -164:</p>

17 <p>The long<a>division</a>method is typically used for finding the square roots of positive non-perfect square numbers. Here’s how to find the square root of 164 using the long division method, then apply the result to -164:</p>

19 <p><strong>Step 1:</strong>Group the numbers from right to left. For 164, group as 64 and 1.</p>

18 <p><strong>Step 1:</strong>Group the numbers from right to left. For 164, group as 64 and 1.</p>

20 <p><strong>Step 2:</strong>Find n whose square is 1. Here, n is 1 because 1 x 1 = 1. Subtract 1 from 1, resulting in a<a>remainder</a>of 0.</p>

19 <p><strong>Step 2:</strong>Find n whose square is 1. Here, n is 1 because 1 x 1 = 1. Subtract 1 from 1, resulting in a<a>remainder</a>of 0.</p>

21 <p><strong>Step 3:</strong>Bring down 64, the new<a>dividend</a>. Add the previous<a>divisor</a>1 to itself to get 2, the new divisor.</p>

20 <p><strong>Step 3:</strong>Bring down 64, the new<a>dividend</a>. Add the previous<a>divisor</a>1 to itself to get 2, the new divisor.</p>

22 <p><strong>Step 4:</strong>Find n such that 2n x n ≤ 64. Here, n is 8 because 28 x 8 = 224, and 224 is<a>less than</a>640.</p>

21 <p><strong>Step 4:</strong>Find n such that 2n x n ≤ 64. Here, n is 8 because 28 x 8 = 224, and 224 is<a>less than</a>640.</p>

23 <p><strong>Step 5:</strong>Subtract 224 from 640 to get a remainder of 416 and continue the process to get more<a>decimal</a>places. Once you find the square root of 164, multiply the result by i for -164.</p>

22 <p><strong>Step 5:</strong>Subtract 224 from 640 to get a remainder of 416 and continue the process to get more<a>decimal</a>places. Once you find the square root of 164, multiply the result by i for -164.</p>

24 <h2>Square Root of 164 by Approximation Method</h2>

23 <h2>Square Root of 164 by Approximation Method</h2>

25 <p>The approximation method is another way to find square roots. Here’s how to find the square root of 164 using the approximation method:</p>

24 <p>The approximation method is another way to find square roots. Here’s how to find the square root of 164 using the approximation method:</p>

26 <p><strong>Step 1:</strong>Identify the closest perfect squares to 164. The nearest perfect squares are 144 (12²) and 169 (13²). Thus, √164 is between 12 and 13.</p>

25 <p><strong>Step 1:</strong>Identify the closest perfect squares to 164. The nearest perfect squares are 144 (12²) and 169 (13²). Thus, √164 is between 12 and 13.</p>

27 <p><strong>Step 2:</strong>Use the approximation<a>formula</a>: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Applying it: (164 - 144) / (169 - 144) = 20 / 25 = 0.8 Add the decimal to the lower bound: 12 + 0.8 = 12.8 For -164, multiply the result by i to get i(12.8).</p>

26 <p><strong>Step 2:</strong>Use the approximation<a>formula</a>: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Applying it: (164 - 144) / (169 - 144) = 20 / 25 = 0.8 Add the decimal to the lower bound: 12 + 0.8 = 12.8 For -164, multiply the result by i to get i(12.8).</p>

28 <h2>Common Mistakes and How to Avoid Them in the Square Root of -164</h2>

27 <h2>Common Mistakes and How to Avoid Them in the Square Root of -164</h2>

29 <p>Students often make mistakes while finding square roots, such as forgetting about the imaginary unit when dealing with negative numbers. Let’s explore some common mistakes in detail.</p>

28 <p>Students often make mistakes while finding square roots, such as forgetting about the imaginary unit when dealing with negative numbers. Let’s explore some common mistakes in detail.</p>

30 <h3>Problem 1</h3>

29 <h3>Problem 1</h3>

31 <p>If Max wants to find the result of i times the square root of 138, what will it be?</p>

30 <p>If Max wants to find the result of i times the square root of 138, what will it be?</p>

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

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

33 <p>The result is approximately 11.75i.</p>

32 <p>The result is approximately 11.75i.</p>

34 <h3>Explanation</h3>

33 <h3>Explanation</h3>

35 <p>To find the result, calculate the square root of 138, which is approximately 11.75, and multiply by i.</p>

34 <p>To find the result, calculate the square root of 138, which is approximately 11.75, and multiply by i.</p>

36 <p>Well explained 👍</p>

35 <p>Well explained 👍</p>

37 <h3>Problem 2</h3>

36 <h3>Problem 2</h3>

38 <p>A square has an area of -164 square units. What is the side length in terms of imaginary numbers?</p>

37 <p>A square has an area of -164 square units. What is the side length in terms of imaginary numbers?</p>

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

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

40 <p>The side length is i√164.</p>

39 <p>The side length is i√164.</p>

41 <h3>Explanation</h3>

40 <h3>Explanation</h3>

42 <p>The side length of a square is the square root of its area.</p>

41 <p>The side length of a square is the square root of its area.</p>

43 <p>For negative areas, use i to denote the imaginary part: i√164.</p>

42 <p>For negative areas, use i to denote the imaginary part: i√164.</p>

44 <p>Well explained 👍</p>

43 <p>Well explained 👍</p>

45 <h3>Problem 3</h3>

44 <h3>Problem 3</h3>

46 <p>Calculate i√164 x 3.</p>

45 <p>Calculate i√164 x 3.</p>

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

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

48 <p>The result is approximately 38.4i.</p>

47 <p>The result is approximately 38.4i.</p>

49 <h3>Explanation</h3>

48 <h3>Explanation</h3>

50 <p>First, find the square root of 164, which is approximately 12.8, then multiply by 3 and add the imaginary unit: 12.8 x 3 = 38.4, thus 38.4i.</p>

49 <p>First, find the square root of 164, which is approximately 12.8, then multiply by 3 and add the imaginary unit: 12.8 x 3 = 38.4, thus 38.4i.</p>

51 <p>Well explained 👍</p>

50 <p>Well explained 👍</p>

52 <h3>Problem 4</h3>

51 <h3>Problem 4</h3>

53 <p>What is the square root of (138 - 2) in terms of imaginary numbers?</p>

52 <p>What is the square root of (138 - 2) in terms of imaginary numbers?</p>

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

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

55 <p>The square root is approximately 11.66i.</p>

54 <p>The square root is approximately 11.66i.</p>

56 <h3>Explanation</h3>

55 <h3>Explanation</h3>

57 <p>First, find the square root of (138 - 2) = 136, which is approximately 11.66.</p>

56 <p>First, find the square root of (138 - 2) = 136, which is approximately 11.66.</p>

58 <p>Thus, the square root of the negative is 11.66i.</p>

57 <p>Thus, the square root of the negative is 11.66i.</p>

59 <p>Well explained 👍</p>

58 <p>Well explained 👍</p>

60 <h3>Problem 5</h3>

59 <h3>Problem 5</h3>

61 <p>If a rectangle has a length of i√138 units and a width of 38 units, what is its perimeter in terms of imaginary numbers?</p>

60 <p>If a rectangle has a length of i√138 units and a width of 38 units, what is its perimeter in terms of imaginary numbers?</p>

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

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

63 <p>The perimeter is approximately 77.48 + 11.66i units.</p>

62 <p>The perimeter is approximately 77.48 + 11.66i units.</p>

64 <h3>Explanation</h3>

63 <h3>Explanation</h3>

65 <p>Perimeter of the rectangle = 2 × (length + width), where length = i√138 ≈ 11.66i, width = 38.</p>

64 <p>Perimeter of the rectangle = 2 × (length + width), where length = i√138 ≈ 11.66i, width = 38.</p>

66 <p>Thus, the perimeter = 2 × (11.66i + 38) = 77.48 + 23.32i.</p>

65 <p>Thus, the perimeter = 2 × (11.66i + 38) = 77.48 + 23.32i.</p>

67 <p>Well explained 👍</p>

66 <p>Well explained 👍</p>

68 <h2>FAQ on Square Root of -164</h2>

67 <h2>FAQ on Square Root of -164</h2>

69 <h3>1.What is the square root of -164 in its simplest form?</h3>

68 <h3>1.What is the square root of -164 in its simplest form?</h3>

70 <p>The simplest form of the square root of -164 is i√164, where i is the imaginary unit.</p>

69 <p>The simplest form of the square root of -164 is i√164, where i is the imaginary unit.</p>

71 <h3>2.What are the factors of 164?</h3>

70 <h3>2.What are the factors of 164?</h3>

72 <p>The factors of 164 are 1, 2, 4, 41, 82, and 164.</p>

71 <p>The factors of 164 are 1, 2, 4, 41, 82, and 164.</p>

73 <h3>3.Calculate the square of -164.</h3>

72 <h3>3.Calculate the square of -164.</h3>

74 <p>The square of -164 is 26896, as (-164)² = 26896.</p>

73 <p>The square of -164 is 26896, as (-164)² = 26896.</p>

75 <h3>4.Is 164 a prime number?</h3>

74 <h3>4.Is 164 a prime number?</h3>

76 <p>No, 164 is not a<a>prime number</a>, as it has more than two factors.</p>

75 <p>No, 164 is not a<a>prime number</a>, as it has more than two factors.</p>

77 <h3>5.Is -164 divisible by any numbers?</h3>

76 <h3>5.Is -164 divisible by any numbers?</h3>

78 <p>-164 is divisible by its factors: 1, 2, 4, 41, 82, and 164.</p>

77 <p>-164 is divisible by its factors: 1, 2, 4, 41, 82, and 164.</p>

79 <h2>Important Glossaries for the Square Root of -164</h2>

78 <h2>Important Glossaries for the Square Root of -164</h2>

80 <ul><li><strong>Imaginary Unit:</strong>The imaginary unit i is defined such that i² = -1. It is used to express the square roots of negative numbers.</li>

79 <ul><li><strong>Imaginary Unit:</strong>The imaginary unit i is defined such that i² = -1. It is used to express the square roots of negative numbers.</li>

81 </ul><ul><li><strong>Square Root:</strong>The square root of a number x is a number y such that y² = x. For negative numbers, the square root involves the imaginary unit i.</li>

80 </ul><ul><li><strong>Square Root:</strong>The square root of a number x is a number y such that y² = x. For negative numbers, the square root involves the imaginary unit i.</li>

82 </ul><ul><li><strong>Prime Factorization:</strong>Breaking down a number into the product of its prime factors. For example, 164 = 2² x 41.</li>

81 </ul><ul><li><strong>Prime Factorization:</strong>Breaking down a number into the product of its prime factors. For example, 164 = 2² x 41.</li>

83 </ul><ul><li><strong>Perfect Square:</strong>A number that is the square of an integer. For example, 144 is a perfect square since 12² = 144.</li>

82 </ul><ul><li><strong>Perfect Square:</strong>A number that is the square of an integer. For example, 144 is a perfect square since 12² = 144.</li>

84 </ul><ul><li><strong>Approximation:</strong>The process of finding a close estimate of a number, often used for non-perfect squares.</li>

83 </ul><ul><li><strong>Approximation:</strong>The process of finding a close estimate of a number, often used for non-perfect squares.</li>

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

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

86 <p>▶</p>

85 <p>▶</p>

87 <h2>Jaskaran Singh Saluja</h2>

86 <h2>Jaskaran Singh Saluja</h2>

88 <h3>About the Author</h3>

87 <h3>About the Author</h3>

89 <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>

88 <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>

90 <h3>Fun Fact</h3>

89 <h3>Fun Fact</h3>

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

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