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Last updated on March 28th, 2025
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 such as vehicle design, finance, etc. Here, we will discuss the square root of 13500.
The square root is the inverse of the square of the number. 13500 is not a perfect square. The square root of 13500 is expressed in both radical and exponential form. In the radical form, it is expressed as √13500, whereas (13500)(1/2) in exponential form. √13500 ≈ 116.1895, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not ideal for non-perfect square numbers, where long-division and approximation methods are used. Let us now learn the following methods:
The product of prime factors is the prime factorization of a number. Now let us look at how 13500 is broken down into its prime factors.
Step 1: Finding the prime factors of 13500 Breaking it down, we get 2 x 2 x 3 x 3 x 3 x 5 x 5 x 5: 22 x 33 x 53
Step 2: Now we found out the prime factors of 13500. The second step is to make pairs of those prime factors. Since 13500 is not a perfect square, the digits of the number can’t be grouped in perfect pairs.
Therefore, calculating the exact square root of 13500 using prime factorization alone is not possible.
The long division 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 square root using the long division method, step by step.
Step 1: To begin, we need to group the numbers from right to left. In the case of 13500, we need to group it as 500 and 13.
Step 2: Now we need to find n whose square is ≤ 13. We can say n is ‘3’ because 3 x 3 = 9, which is less than or equal to 13. Now the quotient is 3, and after subtracting 9 from 13, the remainder is 4.
Step 3: Now let us bring down 500, which is the new dividend. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we have 6n as the new divisor; we need to find the value of n.
Step 5: The next step is finding 6n x n ≤ 450. Let us consider n as 7, now 67 x 7 = 469.
Step 6: Subtract 450 from 469, and the difference is 19, with a quotient of 37.
Step 7: 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 1900.
Step 8: Now we need to find the new divisor that is 746 because 746 x 2 = 1492.
Step 9: Subtracting 1492 from 1900, we get the result 408.
Step 10: Now the quotient is 116.2.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal values continue till the remainder is zero.
So the square root of √13500 is approximately 116.19.
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 13500 using the approximation method.
Step 1: Now we have to find the closest perfect square of √13500. The smallest perfect square below 13500 is 12996, and the largest perfect square above 13500 is 14400. √13500 falls somewhere between 114 and 120.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)
Using the formula (13500 - 12996) ÷ (14400 - 12996) ≈ 0.1895 Using the formula, we identified the decimal point of our square root.
The next step is adding the value we got initially to the decimal number, which is 116 + 0.1895 ≈ 116.19, so the square root of 13500 is approximately 116.19.
Can you help Max find the area of a square box if its side length is given as √135?
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Calculate √13500 x 5.
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Find the perimeter of the rectangle if its length ‘l’ is √135 units and the width ‘w’ is 35 units.
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.
: He loves to play the quiz with kids through algebra to make kids love it.