Last updated on May 26th, 2025
If a number is multiplied by itself, the result is a square. The inverse of squaring a number is finding its square root. Square roots are used in various fields, including engineering and finance. Here, we will discuss the square root of 652.
The square root is the inverse operation of squaring a number. 652 is not a perfect square. The square root of 652 can be expressed in both radical form and exponential form. In radical form, it is expressed as √652, and in exponential form as \(652^{1/2}\). The value of √652 is approximately 25.529, which is an irrational number as it cannot be expressed as a fraction of two integers.
The prime factorization method is typically used for perfect square numbers. However, for non-perfect squares, methods like long division and approximation are used. Let's explore these methods:
The prime factorization of a number is the product of its prime factors. Let's break down 652 into its prime factors:
Step 1: Finding the prime factors of 652 Breaking it down, we get 2 x 2 x 163: (2^2 times 163^1)
Step 2: Since 652 is not a perfect square, we cannot group all the prime factors into pairs.
Therefore, calculating √652 using prime factorization alone is not feasible.
The long division method is useful for finding the square root of non-perfect squares. Here is how to use this method step-by-step:
Step 1: Group the digits of 652 from right to left as '52' and '6'.
Step 2: Find the largest number whose square is less than or equal to 6. This number is 2, since (2^2 = 4). Subtract 4 from 6, leaving a remainder of 2.
Step 3: Bring down the next pair, 52, to get the new dividend, 252. Double the divisor from step 2 (2), giving us 4, and append a digit to form a new divisor.
Step 4: Determine the largest digit, n, so that 4n x n ≤ 252. Here, n = 5, since 45 x 5 = 225.
Step 5: Subtract 225 from 252, resulting in a remainder of 27. Bring down two zeros to form the new dividend, 2700.
Step 6: Repeat the process until the desired precision is achieved. Our quotient so far is approximately 25.52.
The approximation method provides a quick way to estimate square roots. Here's how to find √652 using approximation:
Step 1: Identify perfect squares nearest to 652. The closest perfect squares are 625 (25²) and 676 (26²). Therefore, √652 is between 25 and 26.
Step 2: Use the formula: ((text{Given number} - text{Lower perfect square}) / (text{Higher perfect square} - text{Lower perfect square})). Calculating, we get: ((652 - 625) / (676 - 625) = 27 / 51 approx 0.529).
Step 3: Add this decimal to the lower square root: 25 + 0.529 = 25.529.
Hence, the approximate value of √652 is 25.529.
Students often make errors while finding square roots, such as neglecting the negative square root or misapplying methods. Let's explore common mistakes in more detail.
Can you help Max find the area of a square box if its side length is given as √652?
The area of the square box is 652 square units.
The area of a square is given by side².
The side length is √652, so the area is (√652)² = 652.
A square-shaped plot measuring 652 square feet is constructed. If each side is √652 feet, what is the area of half the plot?
326 square feet
Since the plot is square-shaped, dividing the total area by 2 gives the area of half the plot.
652 / 2 = 326
So, half of the plot measures 326 square feet.
Calculate √652 x 5.
127.645
First, find the square root of 652, approximately 25.529, then multiply by 5:
25.529 x 5 = 127.645.
What is the square root of (652 + 48)?
26
First, find the sum of 652 + 48 = 700.
The square root of 700 is approximately 26.457, but if rounded to the nearest whole number, it is 26.
Therefore, the square root of 700 is approximately ±26.
Find the perimeter of a rectangle if its length ‘l’ is √652 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 127.058 units.
Perimeter of a rectangle = 2 × (length + width).
Perimeter = 2 × (√652 + 38)
= 2 × (25.529 + 38)
= 2 × 63.529
= 127.058 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.