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2026-01-01
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2026-02-28
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<p>Last updated on<strong>August 5, 2025</strong></p>
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<p>Last updated on<strong>August 5, 2025</strong></p>
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<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, including complex numbers, is used in fields like engineering, physics, and mathematics. Here, we will discuss the square root of -180.</p>
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<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, including complex numbers, is used in fields like engineering, physics, and mathematics. Here, we will discuss the square root of -180.</p>
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<h2>What is the Square Root of -180?</h2>
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<h2>What is the Square Root of -180?</h2>
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<p>The<a>square</a>root is the inverse<a>of</a>the square of a<a>number</a>. However, the square root of a<a>negative number</a>involves<a>complex numbers</a>. The square root of -180 is expressed using the imaginary unit '<a>i</a>'. It is written as √-180 = √180 * i. The value of √180 is approximately 13.416, so √-180 = 13.416i, which is a complex number.</p>
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<p>The<a>square</a>root is the inverse<a>of</a>the square of a<a>number</a>. However, the square root of a<a>negative number</a>involves<a>complex numbers</a>. The square root of -180 is expressed using the imaginary unit '<a>i</a>'. It is written as √-180 = √180 * i. The value of √180 is approximately 13.416, so √-180 = 13.416i, which is a complex number.</p>
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<h2>Finding the Square Root of -180</h2>
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<h2>Finding the Square Root of -180</h2>
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<p>Finding the<a>square root</a>of a negative number requires the introduction of the imaginary unit 'i', where i is defined as √-1. The process involves finding the square root of the positive counterpart first. Let us now learn the following methods:</p>
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<p>Finding the<a>square root</a>of a negative number requires the introduction of the imaginary unit 'i', where i is defined as √-1. The process involves finding the square root of the positive counterpart first. Let us now learn the following methods:</p>
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<p>1. Express the square root of a negative number using i.</p>
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<p>1. Express the square root of a negative number using i.</p>
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<p>2. Calculate the square root of the positive number.</p>
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<p>2. Calculate the square root of the positive number.</p>
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<p>3. Combine the results to express the full square root.</p>
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<p>3. Combine the results to express the full square root.</p>
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<h2>Square Root of -180 by Prime Factorization Method</h2>
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<h2>Square Root of -180 by Prime Factorization Method</h2>
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<p>To express the square root using the<a>prime factorization</a>method, we first find the prime<a>factors</a>of the positive part of the number.</p>
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<p>To express the square root using the<a>prime factorization</a>method, we first find the prime<a>factors</a>of the positive part of the number.</p>
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<p><strong>Step 1:</strong>Finding the prime factors of 180 Breaking it down, we get 2 x 2 x 3 x 3 x 5: 2² x 3² x 5</p>
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<p><strong>Step 1:</strong>Finding the prime factors of 180 Breaking it down, we get 2 x 2 x 3 x 3 x 5: 2² x 3² x 5</p>
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<p><strong>Step 2:</strong>Now we found the prime factors of 180. We express √180 as √(2² x 3² x 5) = 6√5.</p>
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<p><strong>Step 2:</strong>Now we found the prime factors of 180. We express √180 as √(2² x 3² x 5) = 6√5.</p>
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<p><strong>Step 3:</strong>Since the square root involves a negative number, we multiply by i. Thus, √-180 = 6√5 * i.</p>
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<p><strong>Step 3:</strong>Since the square root involves a negative number, we multiply by i. Thus, √-180 = 6√5 * i.</p>
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<h2>Square Root of -180 by Long Division Method</h2>
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<h2>Square Root of -180 by Long Division Method</h2>
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<p>The<a>long division</a>method can be used to find the square root of the positive counterpart, √180, with more precision. However, since -180 is negative, the final result will be multiplied by i.</p>
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<p>The<a>long division</a>method can be used to find the square root of the positive counterpart, √180, with more precision. However, since -180 is negative, the final result will be multiplied by i.</p>
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<p><strong>Step 1:</strong>Group the numbers from right to left. For 180, there's no need for grouping as it's a three-digit number.</p>
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<p><strong>Step 1:</strong>Group the numbers from right to left. For 180, there's no need for grouping as it's a three-digit number.</p>
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<p><strong>Step 2:</strong>Find n such that n² ≤ 1. The closest is 1 since 1² = 1.</p>
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<p><strong>Step 2:</strong>Find n such that n² ≤ 1. The closest is 1 since 1² = 1.</p>
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<p><strong>Step 3:</strong>Bring down the next digits, making the next number 80.</p>
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<p><strong>Step 3:</strong>Bring down the next digits, making the next number 80.</p>
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<p><strong>Step 4:</strong>Double the current<a>quotient</a>(1) to get 2, making 20n the new<a>divisor</a>.</p>
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<p><strong>Step 4:</strong>Double the current<a>quotient</a>(1) to get 2, making 20n the new<a>divisor</a>.</p>
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<p><strong>Step 5:</strong>Find the largest n such that 20n * n ≤ 80. This gives n = 4, so the divisor becomes 24.</p>
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<p><strong>Step 5:</strong>Find the largest n such that 20n * n ≤ 80. This gives n = 4, so the divisor becomes 24.</p>
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<p><strong>Step 6:</strong>Continue this process to find more<a>decimal</a>places, eventually finding √180 ≈ 13.416.</p>
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<p><strong>Step 6:</strong>Continue this process to find more<a>decimal</a>places, eventually finding √180 ≈ 13.416.</p>
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<p><strong>Step 7:</strong>Multiply by i to account for the negative number, so √-180 = 13.416i.</p>
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<p><strong>Step 7:</strong>Multiply by i to account for the negative number, so √-180 = 13.416i.</p>
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<h2>Square Root of -180 by Approximation Method</h2>
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<h2>Square Root of -180 by Approximation Method</h2>
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<p>The approximation method can help in estimating the square root by considering nearby<a>perfect squares</a>.</p>
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<p>The approximation method can help in estimating the square root by considering nearby<a>perfect squares</a>.</p>
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<p><strong>Step 1:</strong>Identify the perfect squares around 180. The perfect squares are 169 (13²) and 196 (14²).</p>
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<p><strong>Step 1:</strong>Identify the perfect squares around 180. The perfect squares are 169 (13²) and 196 (14²).</p>
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<p><strong>Step 2:</strong>√180 lies between 13 and 14. Calculate the approximate decimal using linear interpolation: (180 - 169) / (196 - 169) ≈ 0.407</p>
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<p><strong>Step 2:</strong>√180 lies between 13 and 14. Calculate the approximate decimal using linear interpolation: (180 - 169) / (196 - 169) ≈ 0.407</p>
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<p><strong>Step 3:</strong>Thus, √180 ≈ 13 + 0.407 = 13.407. Multiply this by i to get √-180 ≈ 13.407i.</p>
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<p><strong>Step 3:</strong>Thus, √180 ≈ 13 + 0.407 = 13.407. Multiply this by i to get √-180 ≈ 13.407i.</p>
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<h2>Common Mistakes and How to Avoid Them in the Square Root of -180</h2>
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<h2>Common Mistakes and How to Avoid Them in the Square Root of -180</h2>
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<p>Students often make mistakes when dealing with negative square roots, such as forgetting to include the imaginary unit 'i'. Let's explore some common errors and how to avoid them.</p>
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<p>Students often make mistakes when dealing with negative square roots, such as forgetting to include the imaginary unit 'i'. Let's explore some common errors and how to avoid them.</p>
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<h3>Problem 1</h3>
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<h3>Problem 1</h3>
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<p>If Max has a complex number 3 + √-180, what is its modulus?</p>
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<p>If Max has a complex number 3 + √-180, what is its modulus?</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The modulus is approximately 13.76.</p>
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<p>The modulus is approximately 13.76.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>The modulus of a complex number a + bi is √(a² + b²). For 3 + 13.416i, the modulus is √(3² + 13.416²) ≈ 13.76.</p>
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<p>The modulus of a complex number a + bi is √(a² + b²). For 3 + 13.416i, the modulus is √(3² + 13.416²) ≈ 13.76.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 2</h3>
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<h3>Problem 2</h3>
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<p>A rectangle has a length of √-180 and a width of 5. What is its area?</p>
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<p>A rectangle has a length of √-180 and a width of 5. What is its area?</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The area is a complex number: 67.08i square units.</p>
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<p>The area is a complex number: 67.08i square units.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Area = length × width.</p>
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<p>Area = length × width.</p>
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<p>Length = √-180 = 13.416i, Width = 5.</p>
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<p>Length = √-180 = 13.416i, Width = 5.</p>
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<p>Area = 13.416i × 5 = 67.08i square units.</p>
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<p>Area = 13.416i × 5 = 67.08i square units.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 3</h3>
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<h3>Problem 3</h3>
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<p>Calculate 2 × √-180.</p>
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<p>Calculate 2 × √-180.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The result is approximately 26.832i.</p>
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<p>The result is approximately 26.832i.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>First, find √-180 ≈ 13.416i.</p>
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<p>First, find √-180 ≈ 13.416i.</p>
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<p>Then, multiply by 2: 2 × 13.416i = 26.832i.</p>
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<p>Then, multiply by 2: 2 × 13.416i = 26.832i.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 4</h3>
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<h3>Problem 4</h3>
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<p>What is the square root of (-180 + 20) in terms of i?</p>
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<p>What is the square root of (-180 + 20) in terms of i?</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The square root is approximately 12.806i.</p>
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<p>The square root is approximately 12.806i.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>First, find the sum: -180 + 20 = -160. Then, √-160 = √160 * i ≈ 12.806i.</p>
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<p>First, find the sum: -180 + 20 = -160. Then, √-160 = √160 * i ≈ 12.806i.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 5</h3>
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<h3>Problem 5</h3>
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<p>Find the hypotenuse of a right triangle with legs of 12 and √-180.</p>
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<p>Find the hypotenuse of a right triangle with legs of 12 and √-180.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The hypotenuse is a complex number: approximately 18.44i.</p>
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<p>The hypotenuse is a complex number: approximately 18.44i.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Hypotenuse, h = √(12² + (13.416i)²).</p>
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<p>Hypotenuse, h = √(12² + (13.416i)²).</p>
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<p>Since (13.416i)² is negative, the hypotenuse will involve complex arithmetic: h ≈ 18.44i.</p>
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<p>Since (13.416i)² is negative, the hypotenuse will involve complex arithmetic: h ≈ 18.44i.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h2>FAQ on Square Root of -180</h2>
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<h2>FAQ on Square Root of -180</h2>
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<h3>1.What is √-180 in its simplest form?</h3>
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<h3>1.What is √-180 in its simplest form?</h3>
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<p>√-180 is expressed in its simplest form as 6√5 * i, where i is the imaginary unit.</p>
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<p>√-180 is expressed in its simplest form as 6√5 * i, where i is the imaginary unit.</p>
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<h3>2.What are the factors of 180?</h3>
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<h3>2.What are the factors of 180?</h3>
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<p>Factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.</p>
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<p>Factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.</p>
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<h3>3.Calculate the square of 180.</h3>
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<h3>3.Calculate the square of 180.</h3>
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<p>The square of 180 is 180 x 180 = 32400.</p>
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<p>The square of 180 is 180 x 180 = 32400.</p>
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<h3>4.Is 180 a prime number?</h3>
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<h3>4.Is 180 a prime number?</h3>
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<h3>5.What is the imaginary unit 'i'?</h3>
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<h3>5.What is the imaginary unit 'i'?</h3>
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<p>The imaginary unit 'i' is defined as √-1, used to represent the square root of negative numbers.</p>
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<p>The imaginary unit 'i' is defined as √-1, used to represent the square root of negative numbers.</p>
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<h2>Important Glossaries for the Square Root of -180</h2>
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<h2>Important Glossaries for the Square Root of -180</h2>
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<ul><li><strong>Square root:</strong>The square root is the inverse operation of squaring a number. For negative numbers, it involves the imaginary unit 'i'.</li>
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<ul><li><strong>Square root:</strong>The square root is the inverse operation of squaring a number. For negative numbers, it involves the imaginary unit 'i'.</li>
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</ul><ul><li><strong>Imaginary unit:</strong>Denoted by 'i', it is defined as √-1 and is used to express the square root of negative numbers.</li>
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</ul><ul><li><strong>Imaginary unit:</strong>Denoted by 'i', it is defined as √-1 and is used to express the square root of negative numbers.</li>
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</ul><ul><li><strong>Complex number:</strong>A number that comprises a real part and an imaginary part, usually expressed in the form a + bi.</li>
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</ul><ul><li><strong>Complex number:</strong>A number that comprises a real part and an imaginary part, usually expressed in the form a + bi.</li>
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</ul><ul><li><strong>Modulus of a complex number:</strong>The modulus is the magnitude of a complex number, calculated as √(a² + b²) for a complex number a + bi.</li>
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</ul><ul><li><strong>Modulus of a complex number:</strong>The modulus is the magnitude of a complex number, calculated as √(a² + b²) for a complex number a + bi.</li>
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</ul><ul><li><strong>Prime factorization:</strong>The process of breaking down a number into its prime factors, which are multiplied to give the original number.</li>
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</ul><ul><li><strong>Prime factorization:</strong>The process of breaking down a number into its prime factors, which are multiplied to give the original number.</li>
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</ul><p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>
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</ul><p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>
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<h2>Jaskaran Singh Saluja</h2>
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<h2>Jaskaran Singh Saluja</h2>
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<h3>About the Author</h3>
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<h3>About the Author</h3>
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<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>
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<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>
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<h3>Fun Fact</h3>
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<h3>Fun Fact</h3>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>