Last updated on May 26th, 2025
If a number is multiplied by itself, the result is a square. The inverse operation is finding the square root. Square roots are used in fields like vehicle design and finance. Here, we will discuss the square root of 1728.
The square root is the inverse of squaring a number. 1728 is a perfect square. The square root of 1728 is expressed in both radical and exponential form. In radical form, it is expressed as √1728, whereas (1728)^(1/2) is its exponential form. √1728 = 41.569219, which is a rational number because it can be expressed as an integer.
The prime factorization method is used for perfect square numbers. For non-perfect squares, methods like long division and approximation are used. Let's now learn the following methods:
The prime factorization of a number is the product of its prime factors. Let's see how 1728 is broken down into its prime factors.
Step 1: Finding the prime factors of 1728 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3: 2^6 x 3^3
Step 2: The prime factors of 1728 can be grouped in pairs, allowing us to find the square root using prime factorization. √1728 = (2^6 x 3^3)^(1/2) = 2^3 x 3^(3/2) = 12 x 3 = 36
The long division method is used for both perfect and non-perfect square numbers. Here's how to find the square root using the long division method step by step:
Step 1: Begin by grouping the numbers from right to left. In the case of 1728, we group it as 17 and 28.
Step 2: Find n whose square is less than or equal to 17. We choose n as 4 because 4 x 4 = 16. The quotient is 4 and the remainder is 1.
Step 3: Bring down 28 to get a new dividend of 128. Add the old divisor with the same number, 4 + 4 = 8, which will be our new divisor.
Step 4: Find n such that 8n x n ≤ 128. If n = 1, then 81 x 1 = 81.
Step 5: Subtract 81 from 128 to get a remainder of 47. The quotient is 41.
Step 6: Bring down two zeros to form 4700 as the new dividend.
Step 7: Find a new divisor that results in 419 x 9 = 3771.
Step 8: Subtract 3771 from 4700 to get 929. The quotient is now 41.9.
Step 9: Continue these steps until you achieve the desired decimal precision.
Thus, √1728 ≈ 41.57.
The approximation method is another way to find square roots. Let's see how to find the square root of 1728 using the approximation method:
Step 1: Identify the closest perfect squares to √1728.
The closest perfect square less than 1728 is 1600, and the closest perfect square greater than 1728 is 1764.
√1728 falls between 40 and 42.
Step 2: Apply the formula:
(Given number - closest smaller perfect square) ÷ (closest larger perfect square - closest smaller perfect square).
(1728 - 1600) ÷ (1764 - 1600) = 128 ÷ 164 = 0.78048
Add the result to the smaller square root: 40 + 0.78048 ≈ 40.78
Students often make mistakes when finding square roots, such as forgetting about the negative square root or skipping steps in the long division method. Let's review a few common mistakes.
Can you help Sam find the area of a square box if its side length is √1728?
The area of the square is 1728 square units.
The area of the square = side^2. The side length is given as √1728. Area = (√1728)^2 = 1728. Therefore, the area of the square box is 1728 square units.
A square-shaped land measuring 1728 square feet is constructed; if each of the sides is √1728, what will be the square feet of half of the land?
864 square feet
Since the land is square-shaped, we can divide the given area by 2. 1728 ÷ 2 = 864 So, half of the land measures 864 square feet.
Calculate √1728 x 5.
207.85
First, find the square root of 1728, which is approximately 41.57. Then multiply 41.57 by 5. 41.57 x 5 = 207.85
What will be the square root of (1600 + 128)?
The square root is 42.
To find the square root, calculate the sum (1600 + 128). 1600 + 128 = 1728, and √1728 = 41.57, rounded to 42. Therefore, the square root of (1600 + 128) is approximately ±42.
Find the perimeter of a rectangle if its length ‘l’ is √1728 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 159.14 units.
Perimeter of the rectangle = 2 × (length + width) Perimeter = 2 × (√1728 + 38) = 2 × (41.57 + 38) = 2 × 79.57 = 159.14 units.
Square root: The square root is the inverse operation of squaring a number. For example, 6^2 = 36, and the inverse is √36 = 6. Rational number: A rational number is a number that can be expressed as a fraction of two integers, where the denominator is not zero. Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6^2. Prime factorization: Prime factorization is expressing a number as the product of its prime factors. For example, the prime factorization of 28 is 2^2 x 7. Long division method: A procedure used to find the square root of a number by dividing and averaging, often used for non-perfect squares.
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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