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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 square root is used in fields like engineering, physics, and complex analysis. Here, we will discuss the square root of -784.</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 square root is used in fields like engineering, physics, and complex analysis. Here, we will discuss the square root of -784.</p>
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<h2>What is the Square Root of -784?</h2>
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<h2>What is the Square Root of -784?</h2>
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<p>The<a>square</a>root is the inverse<a>of</a>the square of a<a>number</a>. Since -784 is negative, its square root is not a<a>real number</a>. Instead, it is an<a>imaginary number</a>. The square root of -784 is expressed in<a>terms</a>of the imaginary unit,<a>i</a>, which is defined as √-1. Thus, the square root of -784 is written as √-784 = √784 * √-1 = 28i. Because it involves i, it is an imaginary number and not a real or rational 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>. Since -784 is negative, its square root is not a<a>real number</a>. Instead, it is an<a>imaginary number</a>. The square root of -784 is expressed in<a>terms</a>of the imaginary unit,<a>i</a>, which is defined as √-1. Thus, the square root of -784 is written as √-784 = √784 * √-1 = 28i. Because it involves i, it is an imaginary number and not a real or rational number.</p>
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<h2>Finding the Square Root of -784</h2>
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<h2>Finding the Square Root of -784</h2>
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<h2>Square Root of -784 by Prime Factorization Method</h2>
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<h2>Square Root of -784 by Prime Factorization Method</h2>
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<p>Since -784 is negative, we consider the prime factorization of its absolute value, 784. Here's how it's done:</p>
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<p>Since -784 is negative, we consider the prime factorization of its absolute value, 784. Here's how it's done:</p>
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<p><strong>Step 1:</strong>Find the prime<a>factors</a>of 784.</p>
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<p><strong>Step 1:</strong>Find the prime<a>factors</a>of 784.</p>
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<p>Breaking it down, we have 2 × 2 × 2 × 2 × 7 × 7, which simplifies to 2⁴ × 7².</p>
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<p>Breaking it down, we have 2 × 2 × 2 × 2 × 7 × 7, which simplifies to 2⁴ × 7².</p>
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<p><strong>Step 2:</strong>Pair the prime factors. The pairings are (2²)² and (7¹)², making 784 a perfect square.</p>
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<p><strong>Step 2:</strong>Pair the prime factors. The pairings are (2²)² and (7¹)², making 784 a perfect square.</p>
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<p><strong>Step 3:</strong>Combine the pairs to find the square root, which is 28.</p>
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<p><strong>Step 3:</strong>Combine the pairs to find the square root, which is 28.</p>
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<p><strong>Step 4:</strong>Attach the imaginary unit i to account for the negative sign, resulting in √-784 = 28i.</p>
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<p><strong>Step 4:</strong>Attach the imaginary unit i to account for the negative sign, resulting in √-784 = 28i.</p>
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<h2>Square Root of -784 by Long Division Method</h2>
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<h2>Square Root of -784 by Long Division Method</h2>
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<p>The<a>long division</a>method is typically used for non-perfect square numbers. Here, we use it to confirm the real part, then apply the imaginary unit.</p>
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<p>The<a>long division</a>method is typically used for non-perfect square numbers. Here, we use it to confirm the real part, then apply the imaginary unit.</p>
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<p><strong>Step 1:</strong>Start with the absolute value, 784, and find its square root, using long division if needed.</p>
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<p><strong>Step 1:</strong>Start with the absolute value, 784, and find its square root, using long division if needed.</p>
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<p><strong>Step 2:</strong>For 784, the perfect square is 28.</p>
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<p><strong>Step 2:</strong>For 784, the perfect square is 28.</p>
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<p><strong>Step 3:</strong>Finally, apply the imaginary unit: √-784 = 28i.</p>
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<p><strong>Step 3:</strong>Finally, apply the imaginary unit: √-784 = 28i.</p>
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<h2>Square Root of -784 by Approximation Method</h2>
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<h2>Square Root of -784 by Approximation Method</h2>
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<p>This method is useful if precise calculation of a non-perfect square is needed, but 784 is a perfect square. Thus, we can directly calculate:</p>
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<p>This method is useful if precise calculation of a non-perfect square is needed, but 784 is a perfect square. Thus, we can directly calculate:</p>
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<p><strong>Step 1:</strong>Determine the nearest perfect squares around 784, but since 784 is a perfect square, √784 = 28.</p>
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<p><strong>Step 1:</strong>Determine the nearest perfect squares around 784, but since 784 is a perfect square, √784 = 28.</p>
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<p><strong>Step 2:</strong>Include the imaginary unit to account for the negative sign: √-784 = 28i.</p>
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<p><strong>Step 2:</strong>Include the imaginary unit to account for the negative sign: √-784 = 28i.</p>
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<h2>Common Mistakes and How to Avoid Them in the Square Root of -784</h2>
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<h2>Common Mistakes and How to Avoid Them in the Square Root of -784</h2>
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<p>Students often make mistakes with negative numbers and imaginary units. Understanding these concepts is crucial. Let's explore some common mistakes and how to avoid them.</p>
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<p>Students often make mistakes with negative numbers and imaginary units. Understanding these concepts is crucial. Let's explore some common mistakes 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>Can you help Max find the area of a square box if its side length is given as √-196?</p>
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<p>Can you help Max find the area of a square box if its side length is given as √-196?</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 of the square is -196 square units, noting it involves imaginary numbers.</p>
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<p>The area of the square is -196 square units, noting it involves imaginary numbers.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>The area of the square = side².</p>
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<p>The area of the square = side².</p>
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<p>The side length is given as √-196 = 14i.</p>
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<p>The side length is given as √-196 = 14i.</p>
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<p>Area of the square = (14i)² = 196i² = -196 (since i² = -1).</p>
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<p>Area of the square = (14i)² = 196i² = -196 (since i² = -1).</p>
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<p>Therefore, the area of the square box is -196 square units.</p>
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<p>Therefore, the area of the square box is -196 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 2</h3>
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<h3>Problem 2</h3>
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<p>A square-shaped plot measuring -784 square feet is envisaged; if each of the sides is √-784, what will be the square feet of half of the plot?</p>
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<p>A square-shaped plot measuring -784 square feet is envisaged; if each of the sides is √-784, what will be the square feet of half of the plot?</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>-392 square feet</p>
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<p>-392 square feet</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Divide the imaginary area by 2, but remember the imaginary nature.</p>
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<p>Divide the imaginary area by 2, but remember the imaginary nature.</p>
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<p>Dividing -784 by 2 gives -392.</p>
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<p>Dividing -784 by 2 gives -392.</p>
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<p>So, half of the plot measures -392 square feet, considering the imaginary aspect.</p>
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<p>So, half of the plot measures -392 square feet, considering the imaginary aspect.</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 √-784 x 5.</p>
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<p>Calculate √-784 x 5.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>140i</p>
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<p>140i</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>The first step is to find the square root of -784, which is 28i, then multiply by 5.</p>
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<p>The first step is to find the square root of -784, which is 28i, then multiply by 5.</p>
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<p>So, 28i x 5 = 140i.</p>
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<p>So, 28i x 5 = 140i.</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 will be the square root of (-400 + 16)?</p>
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<p>What will be the square root of (-400 + 16)?</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 ±20i</p>
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<p>The square root is ±20i</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>To find the square root, calculate (-400 + 16) = -384.</p>
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<p>To find the square root, calculate (-400 + 16) = -384.</p>
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<p>Then, √-384 = √384 * i = 20i (assuming simplification though √384 isn't perfect).</p>
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<p>Then, √-384 = √384 * i = 20i (assuming simplification though √384 isn't perfect).</p>
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<p>Therefore, the square root of (-400 + 16) is ±20i.</p>
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<p>Therefore, the square root of (-400 + 16) is ±20i.</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 perimeter of the square if its side ‘s’ is √-784 units.</p>
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<p>Find the perimeter of the square if its side ‘s’ is √-784 units.</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 perimeter is 112i units.</p>
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<p>The perimeter is 112i units.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Perimeter of a square = 4 × side</p>
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<p>Perimeter of a square = 4 × side</p>
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<p>Perimeter = 4 × √-784 = 4 × 28i = 112i units.</p>
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<p>Perimeter = 4 × √-784 = 4 × 28i = 112i units.</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 -784</h2>
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<h2>FAQ on Square Root of -784</h2>
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<h3>1.What is √-784 in its simplest form?</h3>
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<h3>1.What is √-784 in its simplest form?</h3>
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<p>The simplest form of √-784 is 28i, using the prime factorization 2⁴ × 7² for 784 and the imaginary unit i.</p>
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<p>The simplest form of √-784 is 28i, using the prime factorization 2⁴ × 7² for 784 and the imaginary unit i.</p>
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<h3>2.Is -784 a perfect square?</h3>
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<h3>2.Is -784 a perfect square?</h3>
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<p>Yes, 784 is a perfect square, but -784 is not a real perfect square since it involves imaginary numbers.</p>
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<p>Yes, 784 is a perfect square, but -784 is not a real perfect square since it involves imaginary numbers.</p>
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<h3>3.Calculate the square of 784.</h3>
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<h3>3.Calculate the square of 784.</h3>
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<p>The square of 784 is 784 x 784 = 614656.</p>
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<p>The square of 784 is 784 x 784 = 614656.</p>
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<h3>4.What is the imaginary unit?</h3>
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<h3>4.What is the imaginary unit?</h3>
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<p>The imaginary unit, denoted as i, is defined as the square root of -1. It is used in<a>complex numbers</a>.</p>
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<p>The imaginary unit, denoted as i, is defined as the square root of -1. It is used in<a>complex numbers</a>.</p>
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<h3>5.How do you express -784 as a product of prime factors?</h3>
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<h3>5.How do you express -784 as a product of prime factors?</h3>
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<p>The prime factorization of 784 is 2⁴ × 7². We don't factor -1 but use it to introduce the imaginary unit i.</p>
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<p>The prime factorization of 784 is 2⁴ × 7². We don't factor -1 but use it to introduce the imaginary unit i.</p>
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<h2>Important Glossaries for the Square Root of -784</h2>
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<h2>Important Glossaries for the Square Root of -784</h2>
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<ul><li><strong>Imaginary Number:</strong>A number that involves the imaginary unit i, representing the square root of negative numbers. Example: √-9 = 3i.</li>
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<ul><li><strong>Imaginary Number:</strong>A number that involves the imaginary unit i, representing the square root of negative numbers. Example: √-9 = 3i.</li>
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</ul><ul><li><strong>Complex Number:</strong>A combination of real and imaginary numbers in the form a + bi, where a and b are real numbers.</li>
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</ul><ul><li><strong>Complex Number:</strong>A combination of real and imaginary numbers in the form a + bi, where a and b are real numbers.</li>
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</ul><ul><li><strong>Perfect Square:</strong>A number that is the square of an integer. Example: 784 is a perfect square because 28² = 784.</li>
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</ul><ul><li><strong>Perfect Square:</strong>A number that is the square of an integer. Example: 784 is a perfect square because 28² = 784.</li>
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</ul><ul><li><strong>Imaginary Unit (i):</strong>Defined as √-1, it is used to express complex numbers involving square roots of negative numbers.</li>
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</ul><ul><li><strong>Imaginary Unit (i):</strong>Defined as √-1, it is used to express complex numbers involving square roots of negative numbers.</li>
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</ul><ul><li><strong>Square Root:</strong>The inverse operation of squaring a number, involving both real and imaginary roots for negative numbers.</li>
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</ul><ul><li><strong>Square Root:</strong>The inverse operation of squaring a number, involving both real and imaginary roots for negative numbers.</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>