Last updated on May 26th, 2025
If a number is multiplied by itself, the result is a square. The inverse operation of squaring a number is finding its square root. The square root is used in fields such as vehicle design and finance. Here, we will discuss the square root of 525.
The square root is the inverse of squaring a number. 525 is not a perfect square. The square root of 525 can be expressed in both radical and exponential forms. In radical form, it is expressed as √525, whereas in exponential form, it is (525)(1/2). √525 ≈ 22.91288, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
The prime factorization method is used for perfect squares. However, for non-perfect squares, methods like the long division method and approximation method are used. Let us explore these methods: -
Prime factorization involves expressing a number as a product of its prime factors. Let us break down 525 into its prime factors:
Step 1: Find the prime factors of 525 Breaking it down, we get 3 × 5 × 5 × 7: 31 × 52 × 71
Step 2: Since 525 is not a perfect square, the digits cannot be grouped into pairs.
Therefore, calculating the square root of 525 using prime factorization alone is not feasible.
The long division method is useful for finding the square root of non-perfect squares. Let's find the square root of 525 using this method, step by step:
Step 1: Group the digits of 525 from right to left as 25 and 5.
Step 2: Find the largest integer n whose square is less than or equal to 5. Here, n is 2 because 2 × 2 = 4. The remainder is 1.
Step 3: Bring down the next pair, 25, making the new dividend 125. Add the previous divisor to itself to get 4, making the new divisor.
Step 4: Determine n such that 4n × n ≤ 125. If n is 2, then 42 × 2 = 84.
Step 5: Subtract 84 from 125, giving a remainder of 41. The quotient so far is 22. Step 6: Bring down two zeros, making the new dividend 4100.
Step 7: Find the new divisor, which is 229, because 229 × 9 = 2061.
Step 8: Subtract 2061 from 4100, giving a remainder of 2039.
Step 9: The quotient so far is 22.9. Repeat these steps until you have sufficient decimal places.
Thus, √525 ≈ 22.91.
The approximation method is another approach to find square roots. Let's approximate the square root of 525:
Step 1: Identify the perfect squares closest to 525. 484 and 529 are the nearest perfect squares, with square roots 22 and 23, respectively.
Step 2: Apply the formula:
(Given number - smaller perfect square) / (Larger perfect square - smaller perfect square)
(525 - 484) / (529 - 484) = 41 / 45 ≈ 0.9111
Add this decimal to the smaller root: 22 + 0.9111 = 22.9111
Therefore, √525 ≈ 22.9111.
Mistakes are common when calculating square roots, such as forgetting negative roots or misapplying methods. Let's examine some common errors and how to avoid them.
Can you help Max find the area of a square box if its side length is given as √525?
The area of the square is 525 square units.
The area of a square = side².
The side length is given as √525.
Area = (√525)² = 525.
Therefore, the area of the square box is 525 square units.
A square-shaped building measuring 525 square feet is built; if each side is √525, what will be the square feet of half of the building?
262.5 square feet
Divide the total area by 2 for half of the building: 525 / 2 = 262.5
So half of the building measures 262.5 square feet.
Calculate √525 × 5.
114.56
First, find the square root of 525, which is approximately 22.91288.
Then multiply 22.91288 by 5:
22.91288 × 5 ≈ 114.56
What will be the square root of (525 + 100)?
The square root is approximately 25.
First, find the sum of (525 + 100): 525 + 100 = 625.
Then find the square root: √625 = 25.
Therefore, the square root of (525 + 100) is ±25.
Find the perimeter of a rectangle if its length ‘l’ is √525 units and the width ‘w’ is 50 units.
The perimeter of the rectangle is approximately 145.83 units.
Perimeter of a rectangle = 2 × (length + width).
Perimeter = 2 × (√525 + 50)
= 2 × (22.91288 + 50) ≈ 145.83 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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