Last updated on May 26th, 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 1515.
The square root is the inverse of the square of the number. 1515 is not a perfect square. The square root of 1515 is expressed in both radical and exponential form. In the radical form, it is expressed as √1515, whereas (1515)^(1/2) in the exponential form. √1515 ≈ 38.923, 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 used for non-perfect square numbers where long-division method and approximation method 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 1515 is broken down into its prime factors.
Step 1: Finding the prime factors of 1515 Breaking it down, we get 3 × 5 × 101: 3¹ × 5¹ × 101¹
Step 2: Now we found out the prime factors of 1515. The second step is to make pairs of those prime factors. Since 1515 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 1515 using prime factorization is not possible in terms of a perfect square.
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 with, we need to group the numbers from right to left. In the case of 1515, we need to group it as 15 and 15.
Step 2: Now we need to find n whose square is less than or equal to the first group, which is 15. We can say n as ‘3’ because 3 × 3 = 9 is less than 15. Now the quotient is 3, after subtracting 9 from 15, the remainder is 6.
Step 3: Now let us bring down the next group, which is 15, making the new dividend 615. Add the old divisor with the same number 3 + 3 = 6, which will be our new divisor.
Step 4: The new divisor will be 60n, where n is our next digit in the quotient.
Step 5: Find n such that 60n × n ≤ 615. Let us consider n as 1. Now 601 × 1 = 601.
Step 6: Subtract 601 from 615, the remainder is 14, and update the quotient to 31.
Step 7: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend, making it 1400.
Step 8: Now find the new divisor, which is 620, because 6202 × 2 = 1240.
Step 9: Subtracting 1240 from 1400 gives 160.
Step 10: The quotient is now 38.92.
Step 11: Continue these steps until you get the desired precision.
So the square root of √1515 ≈ 38.923.
The approximation method is another method for finding square roots; it is an easy method to approximate the square root of a given number. Now let us learn how to approximate the square root of 1515 using this method.
Step 1: Find the closest perfect squares surrounding √1515.
The smallest perfect square less than 1515 is 1444 (38²), and the largest perfect square greater than 1515 is 1521 (39²). √1515 falls between 38 and 39.
Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square).
Using the formula, (1515 - 1444) / (1521 - 1444) = 71 / 77 ≈ 0.922 Adding this decimal to the lesser whole number: 38 + 0.922 = 38.922.
Therefore, the square root of 1515 is approximately 38.923.
Students often make mistakes while finding the square root, such as forgetting about the negative square root, skipping steps in the long division method, etc. Now let us look at a few of these mistakes in detail.
Can you help Max find the area of a square box if its side length is given as √1515?
The area of the square is approximately 1515 square units.
The area of the square = side².
The side length is given as √1515.
Area of the square = side² = √1515 × √1515 = 1515.
Therefore, the area of the square box is approximately 1515 square units.
A square-shaped building measuring 1515 square feet is built; if each of the sides is √1515, what will be the square feet of half of the building?
757.5 square feet
We can divide the given area by 2 as the building is square-shaped.
Dividing 1515 by 2, we get 757.5.
So half of the building measures 757.5 square feet.
Calculate √1515 × 5.
194.615
The first step is to find the square root of 1515, which is approximately 38.923.
The second step is to multiply 38.923 by 5.
So 38.923 × 5 = 194.615.
What will be the square root of (1500 + 15)?
The square root is approximately 38.923.
To find the square root, we need to find the sum of (1500 + 15). 1500 + 15 = 1515, and then √1515 ≈ 38.923.
Therefore, the square root of (1500 + 15) is approximately ±38.923.
Find the perimeter of a rectangle if its length ‘l’ is √1515 units and the width ‘w’ is 15 units.
The perimeter of the rectangle is approximately 107.846 units.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√1515 + 15) = 2 × (38.923 + 15) = 2 × 53.923 = 107.846 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.
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