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. The square root is used in various fields, including vehicle design and finance. Here, we will discuss the square root of 721.
The square root is the inverse of squaring a number. 721 is not a perfect square. The square root of 721 is expressed in both radical and exponential form. In radical form, it is expressed as √721, whereas (721)^(1/2) in exponential form. √721 ≈ 26.85144, 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 typically used for perfect square numbers. However, for non-perfect square numbers, methods like the long division method and approximation method are used. Let us now learn the following methods:
Prime factorization involves expressing a number as the product of its prime factors. However, since 721 is not a perfect square, finding its square root using prime factorization is not straightforward.
Step 1: Finding the prime factors of 721
Breaking it down, we get 7 × 103: 7^1 × 103^1
Step 2: Since 721 is not a perfect square, the digits of the number can’t be grouped in pairs that result in a perfect square. Therefore, calculating √721 using prime factorization is not possible.
The long division method is particularly used for non-perfect square numbers. This method involves estimating the square root of a number by dividing it into pairs of digits. Let us now learn how to find the square root using the long division method, step by step.
Step 1: Group the numbers from right to left. For 721, we have groups 72 and 1.
Step 2: Find a number whose square is less than or equal to 72. Let n be 8, since 8 × 8 = 64 ≤ 72. The quotient is 8, and the remainder is 72 - 64 = 8.
Step 3: Bring down 1, making the new dividend 81. Double the quotient, 8, to get 16, which is part of the new divisor.
Step 4: Find a number x such that (160 + x) × x ≤ 81. Let x be 0, since 160 × 0 = 0 ≤ 81.
Step 5: Since the remainder is still 81, add a decimal point and two zeroes, making the new dividend 8100.
Step 6: Using 1600 as the new divisor, find a digit y such that (1600 + y) × y ≤ 8100. Let y be 5, since 1605 × 5 = 8025 ≤ 8100.
Step 7: Subtract 8025 from 8100 to get a new remainder of 75. The quotient is now 26.85.
Step 8: Continue these steps until the desired precision is reached.
The approximation method is another way to find square roots. This method involves estimating the square root by identifying perfect squares closest to the number. Let us learn how to find the square root of 721 using the approximation method.
Step 1: Identify perfect squares around √721. The closest perfect square less than 721 is 676 (26^2), and the closest perfect square greater than 721 is 729 (27^2). √721 falls between 26 and 27.
Step 2: Use linear interpolation to approximate the square root. Using the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (721 - 676) / (729 - 676) = 45 / 53 ≈ 0.849 Adding this to the base of 26 gives 26 + 0.849 = 26.849, so the approximate square root of 721 is ≈ 26.85.
Students often make mistakes while finding square roots, such as forgetting about the negative square root or skipping steps in the long division method. Let's look at a few common mistakes in detail.
Can you help Alex find the area of a square box if its side length is given as √721?
The area of the square is approximately 721 square units.
The area of a square is calculated by squaring the side length.
The side length is given as √721.
Area = side^2 = √721 × √721 = 721.
Therefore, the area of the square box is approximately 721 square units.
A square-shaped garden measures 721 square feet; if each side is √721, what is the area of half of the garden?
360.5 square feet
To find half the area of the garden, divide the total area by 2.
Half of 721 square feet = 721 / 2 = 360.5 square feet.
Calculate √721 × 5.
Approximately 134.257
First, find the square root of 721, which is approximately 26.851.
Multiply this by 5: 26.851 × 5 ≈ 134.257.
What is the square root of (721 + 4)?
The square root is approximately 27.
First, find the sum of 721 + 4, which is 725.
Since 725 is close to 729, which is 27^2, the square root of 725 is approximately 27.
Find the perimeter of a rectangle if its length 'l' is √721 units and the width 'w' is 50 units.
The perimeter of the rectangle is approximately 153.702 units.
Perimeter of a rectangle = 2 × (length + width).
Perimeter = 2 × (√721 + 50) = 2 × (26.851 + 50) = 153.702 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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