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2026-01-01
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2026-02-28
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<p>297 Learners</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 the field of vehicle design, finance, etc. Here, we will discuss the square root of 357.</p>
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<h2>What is the Square Root of 357?</h2>
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<p>The<a>square</a>root is the inverse of the square of the<a>number</a>. 357 is not a<a>perfect square</a>. The square root of 357 is expressed in both radical and<a>exponential form</a>. In the radical form, it is expressed as √357, whereas (357)^(1/2) in the exponential form. √357 ≈ 18.89444, 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 357</h2>
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<p>The<a>prime factorization</a>method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where 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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<li>Long division method</li>
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<li>Approximation method</li>
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</ul><h2>Square Root of 357 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 357 is broken down into its prime factors:</p>
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<p><strong>Step 1:</strong>Finding the prime factors of 357 Breaking it down, we get 3 x 7 x 17: 3^1 x 7^1 x 17^1</p>
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<p><strong>Step 2:</strong>Now we found out the prime factors of 357. The second step is to make pairs of those prime factors. Since 357 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.</p>
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<p>Therefore, calculating 357 using prime factorization is impossible.</p>
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<h3>Explore Our Programs</h3>
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<h2>Square Root of 357 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 357, we need to group it as 57 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 357, we need to group it as 57 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 as ‘1’ because 1 x 1 is<a>less than</a>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 as ‘1’ because 1 x 1 is<a>less than</a>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 57 which is the new<a>dividend</a>. Add the old<a>divisor</a>with the same number 1 + 1 we get 2 which will be our new divisor.</p>
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<p><strong>Step 3:</strong>Now let us bring down 57 which is the new<a>dividend</a>. Add the old<a>divisor</a>with the same number 1 + 1 we get 2 which will be our new divisor.</p>
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<p><strong>Step 4:</strong>The new divisor will be the sum 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 sum 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 x n ≤ 257. 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 x n ≤ 257. Let us consider n as 8, now 28 x 8 = 224.</p>
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<p><strong>Step 6:</strong>Subtract 257 from 224, the difference is 33, and the quotient is 18.</p>
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<p><strong>Step 6:</strong>Subtract 257 from 224, the difference is 33, 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 3300.</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 3300.</p>
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<p><strong>Step 8:</strong>Now we need to find the new divisor that is 189 because 1899 x 9 = 1701.</p>
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<p><strong>Step 8:</strong>Now we need to find the new divisor that is 189 because 1899 x 9 = 1701.</p>
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<p><strong>Step 9:</strong>Subtracting 1701 from 3300 we get the result 1599.</p>
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<p><strong>Step 9:</strong>Subtracting 1701 from 3300 we get the result 1599.</p>
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<p><strong>Step 10:</strong>Now the quotient is 18.9.</p>
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<p><strong>Step 10:</strong>Now the quotient is 18.9.</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 √357 is approximately 18.89.</p>
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<p>So the square root of √357 is approximately 18.89.</p>
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<h2>Square Root of 357 by Approximation Method</h2>
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<p>The approximation method is another method for finding the square roots. 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 357 using the approximation method.</p>
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<p><strong>Step 1:</strong>Now we have to find the closest perfect square of √357.</p>
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<p>The smallest perfect square less than 357 is 324 (18^2) and the largest perfect square<a>greater than</a>357 is 361 (19^2).</p>
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<p>√357 falls somewhere between 18 and 19.</p>
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<p><strong>Step 2:</strong>Now we need to apply the<a>formula</a>that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).</p>
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<p>Going by the formula (357 - 324) / (361 - 324) = 33 / 37 ≈ 0.89.</p>
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<p>Using the formula, we identified the<a>decimal</a>point of our square root.</p>
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<p>The next step is adding the value we got initially to the decimal number which is 18 + 0.89 = 18.89, so the square root of 357 is approximately 18.89.</p>
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<h2>Common Mistakes and How to Avoid Them in the Square Root of 357</h2>
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<p>Students do make mistakes while finding the square root, such as forgetting about the negative square root or 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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<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 √357?</p>
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<p>Okay, lets begin</p>
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<p>The area of the square is approximately 357 square units.</p>
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<h3>Explanation</h3>
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<p>The area of the square = side^2.</p>
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<p>The side length is given as √357.</p>
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<p>Area of the square = side^2 = √357 x √357 ≈ 18.89 × 18.89 ≈ 357</p>
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<p>Therefore, the area of the square box is approximately 357 square units.</p>
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<p>Well explained 👍</p>
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<h3>Problem 2</h3>
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<p>A square-shaped building measuring 357 square feet is built; if each of the sides is √357, 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>Approximately 178.5 square feet.</p>
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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>Dividing 357 by 2, we get approximately 178.5.</p>
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<p>So half of the building measures approximately 178.5 square feet.</p>
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<p>Well explained 👍</p>
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<h3>Problem 3</h3>
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<p>Calculate √357 x 5.</p>
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<p>Okay, lets begin</p>
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<p>Approximately 94.47</p>
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<h3>Explanation</h3>
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<p>The first step is to find the square root of 357 which is approximately 18.89.</p>
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<p>The second step is to multiply 18.89 with 5.</p>
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<p>So 18.89 x 5 ≈ 94.47.</p>
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<p>Well explained 👍</p>
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<h3>Problem 4</h3>
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<p>What will be the square root of (338 + 19)?</p>
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<p>Okay, lets begin</p>
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<p>The square root is 19.</p>
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<h3>Explanation</h3>
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<p>To find the square root, we need to find the sum of (338 + 19).</p>
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<p>338 + 19 = 357, and then √357 ≈ 18.89.</p>
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<p>Therefore, the square root of (338 + 19) is approximately ±18.89.</p>
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<p>Well explained 👍</p>
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<h3>Problem 5</h3>
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<p>Find the perimeter of the rectangle if its length ‘l’ is √357 units and the width ‘w’ is 38 units.</p>
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<p>Okay, lets begin</p>
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<p>The perimeter of the rectangle is approximately 113.78 units.</p>
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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 = 2 × (√357 + 38) ≈ 2 × (18.89 + 38) ≈ 2 × 56.89 ≈ 113.78 units.</p>
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<p>Well explained 👍</p>
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<h2>FAQ on Square Root of 357</h2>
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<h3>1.What is √357 in its simplest form?</h3>
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<p>The prime factorization of 357 is 3 x 7 x 17, so the simplest form of √357 = √(3 x 7 x 17).</p>
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<h3>2.Mention the factors of 357.</h3>
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<p>Factors of 357 are 1, 3, 7, 17, 21, 51, 119, and 357.</p>
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<h3>3.Calculate the square of 357.</h3>
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<p>We get the square of 357 by multiplying the number by itself, that is 357 x 357 = 127,449.</p>
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<h3>4.Is 357 a prime number?</h3>
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<h3>5.357 is divisible by?</h3>
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<p>357 has factors; those are 1, 3, 7, 17, 21, 51, 119, and 357.</p>
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<h2>Important Glossaries for the Square Root of 357</h2>
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<ul><li><strong>Square root:</strong>A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, that is √16 = 4. </li>
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<li><strong>Irrational number:</strong>An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers. </li>
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<li><strong>Principal square root:</strong>A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root. </li>
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<li><strong>Prime factorization:</strong>The process of breaking down a number into its prime factors. For example, the prime factorization of 357 is 3 x 7 x 17. </li>
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<li><strong>Long division method:</strong>A method used to find the square root of non-perfect square numbers, involving systematic division steps to approximate the root.</li>
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</ul><p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>
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<p>▶</p>
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<h2>Jaskaran Singh Saluja</h2>
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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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<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>