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2026-01-01
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2026-02-28
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<p>220 Learners</p>
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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 vehicle design, finance, etc. Here, we will discuss the square root of 312.</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 vehicle design, finance, etc. Here, we will discuss the square root of 312.</p>
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<h2>What is the Square Root of 312?</h2>
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<h2>What is the Square Root of 312?</h2>
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<p>The<a>square</a>root is the inverse of the square of a<a>number</a>. 312 is not a<a>perfect square</a>. The square root of 312 is expressed in both radical and<a>exponential form</a>. In the radical form, it is expressed as √312, whereas 312^(1/2) is the exponential form. √312 ≈ 17.66352, 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>
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<p>The<a>square</a>root is the inverse of the square of a<a>number</a>. 312 is not a<a>perfect square</a>. The square root of 312 is expressed in both radical and<a>exponential form</a>. In the radical form, it is expressed as √312, whereas 312^(1/2) is the exponential form. √312 ≈ 17.66352, 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>
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<h2>Finding the Square Root of 312</h2>
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<h2>Finding the Square Root of 312</h2>
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<p>The<a>prime factorization</a>method is used for perfect square numbers. However, for non-perfect square numbers, the long-<a>division</a>method and approximation method are used. Let us now learn the following methods:</p>
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<p>The<a>prime factorization</a>method is used for perfect square numbers. However, for non-perfect square numbers, the long-<a>division</a>method and approximation method are used. Let us now learn the following methods:</p>
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<ul><li>Prime factorization method</li>
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<ul><li>Prime factorization method</li>
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<li> Long division method</li>
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<li> Long division method</li>
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<li> Approximation method</li>
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<li> Approximation method</li>
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</ul><h2>Square Root of 312 by Prime Factorization Method</h2>
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</ul><h2>Square Root of 312 by Prime Factorization Method</h2>
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<p>The<a>product</a>of prime<a>factors</a>is the prime factorization of a number. Now let us look at how 312 is broken down into its prime factors:</p>
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<p>The<a>product</a>of prime<a>factors</a>is the prime factorization of a number. Now let us look at how 312 is broken down into its prime factors:</p>
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<p><strong>Step 1:</strong>Finding the prime factors of 312 Breaking it down, we get 2 x 2 x 2 x 3 x 13: 2^3 x 3^1 x 13^1</p>
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<p><strong>Step 1:</strong>Finding the prime factors of 312 Breaking it down, we get 2 x 2 x 2 x 3 x 13: 2^3 x 3^1 x 13^1</p>
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<p><strong>Step 2:</strong>Now we have found the prime factors of 312. The second step is to make pairs of those prime factors. Since 312 is not a perfect square, the digits of the number can’t be grouped into pairs.</p>
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<p><strong>Step 2:</strong>Now we have found the prime factors of 312. The second step is to make pairs of those prime factors. Since 312 is not a perfect square, the digits of the number can’t be grouped into pairs.</p>
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<p>Therefore, calculating 312 using prime factorization is not straightforward.</p>
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<p>Therefore, calculating 312 using prime factorization is not straightforward.</p>
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<h2>Square Root of 312 by Long Division Method</h2>
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<h2>Square Root of 312 by Long Division Method</h2>
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<p>The<a>long division</a>method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the<a>square root</a>using the long division method, step by step.</p>
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<p>The<a>long division</a>method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the<a>square root</a>using the long division method, step by step.</p>
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<p><strong>Step 1:</strong>To begin with, we need to group the numbers from right to left. In the case of 312, we group it as 12 and 3.</p>
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<p><strong>Step 1:</strong>To begin with, we need to group the numbers from right to left. In the case of 312, we group it as 12 and 3.</p>
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<p><strong>Step 2:</strong>Now we need to find n whose square is 3. We can say n is '1' because 1^2 is lesser than or equal to 3. Now the<a>quotient</a>is 1; after subtracting 1 from 3, the<a>remainder</a>is 2.</p>
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<p><strong>Step 2:</strong>Now we need to find n whose square is 3. We can say n is '1' because 1^2 is lesser than or equal to 3. Now the<a>quotient</a>is 1; after subtracting 1 from 3, the<a>remainder</a>is 2.</p>
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<p><strong>Step 3:</strong>Now let us bring down 12, which is the new<a>dividend</a>. Add the old<a>divisor</a>with the same number: 1 + 1 = 2, which will be our new divisor.</p>
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<p><strong>Step 3:</strong>Now let us bring down 12, which is the new<a>dividend</a>. Add the old<a>divisor</a>with the same number: 1 + 1 = 2, which will be our new divisor.</p>
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<p><strong>Step 4:</strong>The new divisor will be the<a>sum</a>of the dividend and quotient. Now we get 2n as the new divisor; we need to find the value of n.</p>
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<p><strong>Step 4:</strong>The new divisor will be the<a>sum</a>of the dividend and quotient. Now we get 2n as the new divisor; we need to find the value of n.</p>
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<p><strong>Step 5:</strong>The next step is finding 2n × n ≤ 212. Let us consider n as 8, now 28 x 8 = 224.</p>
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<p><strong>Step 5:</strong>The next step is finding 2n × n ≤ 212. Let us consider n as 8, now 28 x 8 = 224.</p>
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<p><strong>Step 6:</strong>Subtract 212 from 224; the difference is -12, and the quotient is 18.</p>
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<p><strong>Step 6:</strong>Subtract 212 from 224; the difference is -12, and the quotient is 18.</p>
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<p><strong>Step 7:</strong>Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 1200.</p>
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<p><strong>Step 7:</strong>Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 1200.</p>
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<p><strong>Step 8:</strong>Now we need to find the new divisor, which is 176 because 176 x 7 = 1232.</p>
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<p><strong>Step 8:</strong>Now we need to find the new divisor, which is 176 because 176 x 7 = 1232.</p>
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<p><strong>Step 9:</strong>Subtracting 1232 from 1200 gives the result of -32.</p>
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<p><strong>Step 9:</strong>Subtracting 1232 from 1200 gives the result of -32.</p>
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<p><strong>Step 10:</strong>Now the quotient is 17.6.</p>
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<p><strong>Step 10:</strong>Now the quotient is 17.6.</p>
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<p><strong>Step 11:</strong>Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.</p>
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<p><strong>Step 11:</strong>Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.</p>
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<p>So the square root of √312 is approximately 17.66.</p>
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<p>So the square root of √312 is approximately 17.66.</p>
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<h2>Square Root of 312 by Approximation Method</h2>
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<h2>Square Root of 312 by Approximation Method</h2>
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<p>The approximation method is another method for finding square roots, and it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 312 using the approximation method.</p>
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<p>The approximation method is another method for finding square roots, and it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 312 using the approximation method.</p>
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<p><strong>Step 1:</strong>Now we have to find the closest perfect square of √312.</p>
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<p><strong>Step 1:</strong>Now we have to find the closest perfect square of √312.</p>
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<p>The smallest perfect square<a>less than</a>312 is 289, and the largest perfect square<a>greater than</a>312 is 324. √312 falls somewhere between 17 and 18.</p>
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<p>The smallest perfect square<a>less than</a>312 is 289, and the largest perfect square<a>greater than</a>312 is 324. √312 falls somewhere between 17 and 18.</p>
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<p><strong>Step 2:</strong>Now we need to apply the<a>formula</a>: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Applying the formula, (312 - 289) / (324 - 289) = 0.66.</p>
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<p><strong>Step 2:</strong>Now we need to apply the<a>formula</a>: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Applying the formula, (312 - 289) / (324 - 289) = 0.66.</p>
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<p>Using the formula, we identified the<a>decimal</a>point of our square root. The next step is adding the value we got initially to the decimal number, which is 17 + 0.66 = 17.66, so the square root of 312 is approximately 17.66.</p>
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<p>Using the formula, we identified the<a>decimal</a>point of our square root. The next step is adding the value we got initially to the decimal number, which is 17 + 0.66 = 17.66, so the square root of 312 is approximately 17.66.</p>
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<h2>Common Mistakes and How to Avoid Them in the Square Root of 312</h2>
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<h2>Common Mistakes and How to Avoid Them in the Square Root of 312</h2>
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<p>Students make mistakes while finding the square root, like forgetting about the negative square root, skipping long division methods, etc. Now let us look at a few of those mistakes that students tend to make in detail.</p>
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<p>Students make mistakes while finding the square root, like forgetting about the negative square root, skipping long division methods, etc. Now let us look at a few of those mistakes that students tend to make in detail.</p>
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<h2>Download Worksheets</h2>
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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 √312?</p>
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<p>Can you help Max find the area of a square box if its side length is given as √312?</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 approximately 312 square units.</p>
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<p>The area of the square is approximately 312 square units.</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 √312.</p>
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<p>The side length is given as √312.</p>
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<p>Area of the square = side² = √312 × √312 = 312.</p>
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<p>Area of the square = side² = √312 × √312 = 312.</p>
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<p>Therefore, the area of the square box is approximately 312 square units.</p>
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<p>Therefore, the area of the square box is approximately 312 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 building measuring 312 square feet is built; if each of the sides is √312, what will be the square feet of half of the building?</p>
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<p>A square-shaped building measuring 312 square feet is built; if each of the sides is √312, what will be the square feet of half of the building?</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>156 square feet</p>
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<p>156 square feet</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>We can just divide the given area by 2 as the building is square-shaped.</p>
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<p>We can just divide the given area by 2 as the building is square-shaped.</p>
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<p>Dividing 312 by 2, we get 156.</p>
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<p>Dividing 312 by 2, we get 156.</p>
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<p>So half of the building measures 156 square feet.</p>
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<p>So half of the building measures 156 square feet.</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 √312 × 5.</p>
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<p>Calculate √312 × 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>88.32</p>
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<p>88.32</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 312, which is approximately 17.66.</p>
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<p>The first step is to find the square root of 312, which is approximately 17.66.</p>
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<p>The second step is to multiply 17.66 by 5.</p>
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<p>The second step is to multiply 17.66 by 5.</p>
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<p>So 17.66 × 5 = 88.32.</p>
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<p>So 17.66 × 5 = 88.32.</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 (312 + 6)?</p>
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<p>What will be the square root of (312 + 6)?</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 18.</p>
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<p>The square root is 18.</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, we need to find the sum of (312 + 6). 312 + 6 = 318, and then √324 = 18.</p>
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<p>To find the square root, we need to find the sum of (312 + 6). 312 + 6 = 318, and then √324 = 18.</p>
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<p>Therefore, the square root of (312 + 6) is ±18.</p>
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<p>Therefore, the square root of (312 + 6) is ±18.</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 rectangle if its length ‘l’ is √312 units and the width ‘w’ is 38 units.</p>
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<p>Find the perimeter of the rectangle if its length ‘l’ is √312 units and the width ‘w’ is 38 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 of the rectangle is approximately 111.32 units.</p>
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<p>The perimeter of the rectangle is approximately 111.32 units.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Perimeter of the rectangle = 2 × (length + width).</p>
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<p>Perimeter of the rectangle = 2 × (length + width).</p>
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<p>Perimeter = 2 × (√312 + 38) = 2 × (17.66 + 38) = 2 × 55.66 = 111.32 units.</p>
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<p>Perimeter = 2 × (√312 + 38) = 2 × (17.66 + 38) = 2 × 55.66 = 111.32 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 312</h2>
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<h2>FAQ on Square Root of 312</h2>
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<h3>1.What is √312 in its simplest form?</h3>
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<h3>1.What is √312 in its simplest form?</h3>
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<p>The prime factorization of 312 is 2 × 2 × 2 × 3 × 13, so the simplest form of √312 is √(2 × 2 × 2 × 3 × 13).</p>
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<p>The prime factorization of 312 is 2 × 2 × 2 × 3 × 13, so the simplest form of √312 is √(2 × 2 × 2 × 3 × 13).</p>
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<h3>2.Mention the factors of 312.</h3>
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<h3>2.Mention the factors of 312.</h3>
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<p>Factors of 312 are 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, and 312.</p>
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<p>Factors of 312 are 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, and 312.</p>
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<h3>3.Calculate the square of 312.</h3>
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<h3>3.Calculate the square of 312.</h3>
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<p>We get the square of 312 by multiplying the number by itself, that is 312 × 312 = 97344.</p>
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<p>We get the square of 312 by multiplying the number by itself, that is 312 × 312 = 97344.</p>
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<h3>4.Is 312 a prime number?</h3>
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<h3>4.Is 312 a prime number?</h3>
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<h3>5.312 is divisible by?</h3>
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<h3>5.312 is divisible by?</h3>
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<p>312 has many factors; those are 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, and 312.</p>
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<p>312 has many factors; those are 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, and 312.</p>
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<h2>Important Glossaries for the Square Root of 312</h2>
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<h2>Important Glossaries for the Square Root of 312</h2>
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<ul><li><strong>Square root:</strong>A square root is the inverse of a square. Example: 4² = 16 and the inverse of the square is the square root, that is √16 = 4.</li>
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<ul><li><strong>Square root:</strong>A square root is the inverse of a square. Example: 4² = 16 and the inverse of the square is the square root, that is √16 = 4.</li>
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</ul><ul><li><strong>Irrational number:</strong>An irrational number is a number that cannot be written in the form of p/q, q is not equal to zero, and p and q are integers.</li>
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</ul><ul><li><strong>Irrational number:</strong>An irrational number is a number that cannot be written in the form of p/q, q is not equal to zero, and p and q are integers.</li>
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</ul><ul><li><strong>Principal square root:</strong>A number has both positive and negative square roots; however, it is the positive square root that has more prominence due to its uses in the real world. That is why it is also known as the principal square root.</li>
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</ul><ul><li><strong>Principal square root:</strong>A number has both positive and negative square roots; however, it is the positive square root that has more prominence due to its uses in the real world. That is why it is also known as the principal square root.</li>
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</ul><ul><li><strong>Long division method:</strong>A method used to find the square root of non-perfect squares by dividing and averaging.</li>
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</ul><ul><li><strong>Long division method:</strong>A method used to find the square root of non-perfect squares by dividing and averaging.</li>
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</ul><ul><li><strong>Decimal approximation:</strong>The method of estimating the value of a square root to several decimal places for practical use.</li>
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</ul><ul><li><strong>Decimal approximation:</strong>The method of estimating the value of a square root to several decimal places for practical use.</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>