1 added
2 removed
Original
2026-01-01
Modified
2026-02-28
1
-
<p>264 Learners</p>
1
+
<p>300 Learners</p>
2
<p>Last updated on<strong>August 5, 2025</strong></p>
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 concept of square roots extends into the realm of complex numbers when dealing with negative numbers. Here, we will discuss the square root of -108.</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 concept of square roots extends into the realm of complex numbers when dealing with negative numbers. Here, we will discuss the square root of -108.</p>
4
<h2>What is the Square Root of -108?</h2>
4
<h2>What is the Square Root of -108?</h2>
5
<p>The<a>square</a>root is the inverse<a>of</a>the square of a<a>number</a>. When dealing with<a>negative numbers</a>, we use<a>complex numbers</a>to express the square root. The square root of -108 is expressed in<a>terms</a>of the imaginary unit '<a>i</a>', where i = √(-1). Thus, the square root of -108 is written as √(-108) = √(108) * i. The value of √108 is approximately 10.3923, so √(-108) ≈ 10.3923i.</p>
5
<p>The<a>square</a>root is the inverse<a>of</a>the square of a<a>number</a>. When dealing with<a>negative numbers</a>, we use<a>complex numbers</a>to express the square root. The square root of -108 is expressed in<a>terms</a>of the imaginary unit '<a>i</a>', where i = √(-1). Thus, the square root of -108 is written as √(-108) = √(108) * i. The value of √108 is approximately 10.3923, so √(-108) ≈ 10.3923i.</p>
6
<h2>Finding the Square Root of -108</h2>
6
<h2>Finding the Square Root of -108</h2>
7
<p>There are different methods to find square roots, but for negative numbers, the<a>concept of imaginary numbers</a>is used. Here's how to find the<a>square root</a>of -108:</p>
7
<p>There are different methods to find square roots, but for negative numbers, the<a>concept of imaginary numbers</a>is used. Here's how to find the<a>square root</a>of -108:</p>
8
<p>1. Calculate the square root of the positive part, 108.</p>
8
<p>1. Calculate the square root of the positive part, 108.</p>
9
<p>2. Multiply the result by i (the imaginary unit).</p>
9
<p>2. Multiply the result by i (the imaginary unit).</p>
10
<h2>Square Root of -108 Using Imaginary Numbers</h2>
10
<h2>Square Root of -108 Using Imaginary Numbers</h2>
11
<p>To find the square root of -108 using imaginary numbers, we start with the positive part:</p>
11
<p>To find the square root of -108 using imaginary numbers, we start with the positive part:</p>
12
<p><strong>Step 1:</strong>Calculate the square root of 108. The<a>prime factorization</a>of 108 is 2 x 2 x 3 x 3 x 3, or 2² x 3³.</p>
12
<p><strong>Step 1:</strong>Calculate the square root of 108. The<a>prime factorization</a>of 108 is 2 x 2 x 3 x 3 x 3, or 2² x 3³.</p>
13
<p><strong>Step 2:</strong>Simplify √108 = √(2² x 3² x 3) = 2 x 3 x √3 = 6√3 ≈ 10.3923.</p>
13
<p><strong>Step 2:</strong>Simplify √108 = √(2² x 3² x 3) = 2 x 3 x √3 = 6√3 ≈ 10.3923.</p>
14
<p><strong>Step 3:</strong>Multiply by the imaginary unit i: √(-108) = √108 * i = 10.3923i.</p>
14
<p><strong>Step 3:</strong>Multiply by the imaginary unit i: √(-108) = √108 * i = 10.3923i.</p>
15
<h3>Explore Our Programs</h3>
15
<h3>Explore Our Programs</h3>
16
-
<p>No Courses Available</p>
17
<h2>Square Root of -108 by Long Division Method (Imaginary Part)</h2>
16
<h2>Square Root of -108 by Long Division Method (Imaginary Part)</h2>
18
<p>The<a>long division</a>method is typically used for approximating square roots of positive numbers. For -108, we focus on the positive part:</p>
17
<p>The<a>long division</a>method is typically used for approximating square roots of positive numbers. For -108, we focus on the positive part:</p>
19
<p><strong>Step 1:</strong>Use long division to approximate √108, which we already found to be about 10.3923.</p>
18
<p><strong>Step 1:</strong>Use long division to approximate √108, which we already found to be about 10.3923.</p>
20
<p><strong>Step 2:</strong>Multiply this result by i to obtain the square root of the negative number: √(-108) = 10.3923i.</p>
19
<p><strong>Step 2:</strong>Multiply this result by i to obtain the square root of the negative number: √(-108) = 10.3923i.</p>
21
<h2>Square Root of -108 by Approximation Method</h2>
20
<h2>Square Root of -108 by Approximation Method</h2>
22
<p>Approximation helps find the square root of the positive part, 108:</p>
21
<p>Approximation helps find the square root of the positive part, 108:</p>
23
<p><strong>Step 1:</strong>Identify the<a>perfect squares</a>surrounding 108. 100 (10²) and 121 (11²) are the closest perfect squares.</p>
22
<p><strong>Step 1:</strong>Identify the<a>perfect squares</a>surrounding 108. 100 (10²) and 121 (11²) are the closest perfect squares.</p>
24
<p><strong>Step 2:</strong>Approximate between these values: √108 is between 10 and 11. Using the approximation method, we find √108 ≈ 10.3923.</p>
23
<p><strong>Step 2:</strong>Approximate between these values: √108 is between 10 and 11. Using the approximation method, we find √108 ≈ 10.3923.</p>
25
<p><strong>Step 3:</strong>Multiply by i for the imaginary part: √(-108) = 10.3923i.</p>
24
<p><strong>Step 3:</strong>Multiply by i for the imaginary part: √(-108) = 10.3923i.</p>
26
<h2>Common Mistakes and How to Avoid Them in the Square Root of -108</h2>
25
<h2>Common Mistakes and How to Avoid Them in the Square Root of -108</h2>
27
<p>Students often make mistakes when dealing with the square roots of negative numbers, especially with the involvement of imaginary numbers. Let's discuss some common mistakes and how to avoid them.</p>
26
<p>Students often make mistakes when dealing with the square roots of negative numbers, especially with the involvement of imaginary numbers. Let's discuss some common mistakes and how to avoid them.</p>
28
<h3>Problem 1</h3>
27
<h3>Problem 1</h3>
29
<p>Can you help Max find the area of a square box if its side length is given as √(-108)?</p>
28
<p>Can you help Max find the area of a square box if its side length is given as √(-108)?</p>
30
<p>Okay, lets begin</p>
29
<p>Okay, lets begin</p>
31
<p>The area of the square is -108 square units.</p>
30
<p>The area of the square is -108 square units.</p>
32
<h3>Explanation</h3>
31
<h3>Explanation</h3>
33
<p>The area of the square = side².</p>
32
<p>The area of the square = side².</p>
34
<p>The side length is given as √(-108).</p>
33
<p>The side length is given as √(-108).</p>
35
<p>Area of the square = (√(-108))² = -108.</p>
34
<p>Area of the square = (√(-108))² = -108.</p>
36
<p>Therefore, the area of the square box is -108 square units, considering complex units.</p>
35
<p>Therefore, the area of the square box is -108 square units, considering complex units.</p>
37
<p>Well explained 👍</p>
36
<p>Well explained 👍</p>
38
<h3>Problem 2</h3>
37
<h3>Problem 2</h3>
39
<p>A complex square-shaped plot has an area of -108 square meters. What is the length of each side if it's given by √(-108)?</p>
38
<p>A complex square-shaped plot has an area of -108 square meters. What is the length of each side if it's given by √(-108)?</p>
40
<p>Okay, lets begin</p>
39
<p>Okay, lets begin</p>
41
<p>The side length is approximately 10.3923i meters.</p>
40
<p>The side length is approximately 10.3923i meters.</p>
42
<h3>Explanation</h3>
41
<h3>Explanation</h3>
43
<p>The side length of the square is given by the square root of the area. √(-108) = 10.3923i meters, which is the length of each side, considering the imaginary component.</p>
42
<p>The side length of the square is given by the square root of the area. √(-108) = 10.3923i meters, which is the length of each side, considering the imaginary component.</p>
44
<p>Well explained 👍</p>
43
<p>Well explained 👍</p>
45
<h3>Problem 3</h3>
44
<h3>Problem 3</h3>
46
<p>Calculate √(-108) x 5.</p>
45
<p>Calculate √(-108) x 5.</p>
47
<p>Okay, lets begin</p>
46
<p>Okay, lets begin</p>
48
<p>51.9615i</p>
47
<p>51.9615i</p>
49
<h3>Explanation</h3>
48
<h3>Explanation</h3>
50
<p>First, find the square root of -108, which is approximately 10.3923i. Multiply by 5: 10.3923i x 5 = 51.9615i.</p>
49
<p>First, find the square root of -108, which is approximately 10.3923i. Multiply by 5: 10.3923i x 5 = 51.9615i.</p>
51
<p>Well explained 👍</p>
50
<p>Well explained 👍</p>
52
<h3>Problem 4</h3>
51
<h3>Problem 4</h3>
53
<p>What will be the square root of (-100 + 8)?</p>
52
<p>What will be the square root of (-100 + 8)?</p>
54
<p>Okay, lets begin</p>
53
<p>Okay, lets begin</p>
55
<p>The square root is approximately 10.3923i.</p>
54
<p>The square root is approximately 10.3923i.</p>
56
<h3>Explanation</h3>
55
<h3>Explanation</h3>
57
<p>Calculate the sum: (-100 + 8) = -92. Then find the square root: √(-92) = √92 * i ≈ 9.5917i.</p>
56
<p>Calculate the sum: (-100 + 8) = -92. Then find the square root: √(-92) = √92 * i ≈ 9.5917i.</p>
58
<p>Well explained 👍</p>
57
<p>Well explained 👍</p>
59
<h3>Problem 5</h3>
58
<h3>Problem 5</h3>
60
<p>Find the perimeter of a rectangle if its length 'l' is √(-108) units and the width 'w' is 38 units.</p>
59
<p>Find the perimeter of a rectangle if its length 'l' is √(-108) units and the width 'w' is 38 units.</p>
61
<p>Okay, lets begin</p>
60
<p>Okay, lets begin</p>
62
<p>The perimeter is a complex number: 76 + 20.7846i units.</p>
61
<p>The perimeter is a complex number: 76 + 20.7846i units.</p>
63
<h3>Explanation</h3>
62
<h3>Explanation</h3>
64
<p>Perimeter of the rectangle = 2 × (length + width). Perimeter = 2 × (√(-108) + 38) = 2 × (10.3923i + 38). = 76 + 20.7846i units.</p>
63
<p>Perimeter of the rectangle = 2 × (length + width). Perimeter = 2 × (√(-108) + 38) = 2 × (10.3923i + 38). = 76 + 20.7846i units.</p>
65
<p>Well explained 👍</p>
64
<p>Well explained 👍</p>
66
<h2>FAQ on Square Root of -108</h2>
65
<h2>FAQ on Square Root of -108</h2>
67
<h3>1.What is √(-108) in its simplest form?</h3>
66
<h3>1.What is √(-108) in its simplest form?</h3>
68
<p>The prime factorization of 108 is 2² x 3³. Thus, √108 = 6√3. Therefore, in terms of complex numbers, √(-108) = 6√3i.</p>
67
<p>The prime factorization of 108 is 2² x 3³. Thus, √108 = 6√3. Therefore, in terms of complex numbers, √(-108) = 6√3i.</p>
69
<h3>2.What is the principal square root of -108?</h3>
68
<h3>2.What is the principal square root of -108?</h3>
70
<p>The principal square root of -108 is the positive imaginary root, 10.3923i.</p>
69
<p>The principal square root of -108 is the positive imaginary root, 10.3923i.</p>
71
<h3>3.Can the square root of -108 be simplified further?</h3>
70
<h3>3.Can the square root of -108 be simplified further?</h3>
72
<p>The square root of -108 in its simplest form is 6√3i, as it involves a complex number.</p>
71
<p>The square root of -108 in its simplest form is 6√3i, as it involves a complex number.</p>
73
<h3>4.Is -108 a perfect square?</h3>
72
<h3>4.Is -108 a perfect square?</h3>
74
<p>No, -108 is not a perfect square because it is negative, and perfect squares are non-negative<a>integers</a>.</p>
73
<p>No, -108 is not a perfect square because it is negative, and perfect squares are non-negative<a>integers</a>.</p>
75
<h3>5.What does the imaginary unit 'i' represent?</h3>
74
<h3>5.What does the imaginary unit 'i' represent?</h3>
76
<p>The imaginary unit 'i' represents the square root of -1, which is used to express square roots of negative numbers.</p>
75
<p>The imaginary unit 'i' represents the square root of -1, which is used to express square roots of negative numbers.</p>
77
<h2>Important Glossaries for the Square Root of -108</h2>
76
<h2>Important Glossaries for the Square Root of -108</h2>
78
<ul><li><strong>Imaginary Number:</strong>A number that can be written as a real number multiplied by the imaginary unit 'i', which is defined by i² = -1. </li>
77
<ul><li><strong>Imaginary Number:</strong>A number that can be written as a real number multiplied by the imaginary unit 'i', which is defined by i² = -1. </li>
79
<li><strong>Complex Number:</strong>A number that has both a real part and an imaginary part, expressed in the form a + bi, where a and b are real numbers. </li>
78
<li><strong>Complex Number:</strong>A number that has both a real part and an imaginary part, expressed in the form a + bi, where a and b are real numbers. </li>
80
<li><strong>Square Root:</strong>The value that, when multiplied by itself, gives the original number. For negative numbers, it involves the imaginary unit. </li>
79
<li><strong>Square Root:</strong>The value that, when multiplied by itself, gives the original number. For negative numbers, it involves the imaginary unit. </li>
81
<li><strong>Perfect Square:</strong>A number that is the square of an integer. Negative numbers cannot be perfect squares. </li>
80
<li><strong>Perfect Square:</strong>A number that is the square of an integer. Negative numbers cannot be perfect squares. </li>
82
<li><strong>Approximation:</strong>The process of finding a number close to the exact value, often used for irrational numbers or complex calculations.</li>
81
<li><strong>Approximation:</strong>The process of finding a number close to the exact value, often used for irrational numbers or complex calculations.</li>
83
</ul><p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>
82
</ul><p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>
84
<p>▶</p>
83
<p>▶</p>
85
<h2>Jaskaran Singh Saluja</h2>
84
<h2>Jaskaran Singh Saluja</h2>
86
<h3>About the Author</h3>
85
<h3>About the Author</h3>
87
<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>
86
<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
<h3>Fun Fact</h3>
87
<h3>Fun Fact</h3>
89
<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>
88
<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>