Square & Square Root of 72 – Methods, Calculation, Formula, How to find

Square of 72

The square of 72 (72²) is calculated by multiplying 72 by itself:

72×72=5184.

So, the square of 72 is 5184.

Geometrically, if you have a square with each side measuring 72 units, the total area enclosed by the square will be 5184 square units.

Understanding the square of 72 is essential in various mathematical contexts, including geometry, algebra, and arithmetic. It finds applications in calculating areas, volumes, distances, and solving mathematical problems.

Square Root of 72

The square root of 72 (√72) is an irrational number, approximately equal to 8.48528137424.

It represents a number that, when multiplied by itself, results in 72:

72≈8.4852813742472​

Since 72 is not a perfect square, its square root cannot be simplified to a whole number or a simple fraction. Instead, it is a non-repeating, non-terminating decimal.

The square root of 72 finds applications in various fields such as mathematics, physics, engineering, and finance, where precise calculations are required.

Is the Square Root of 72 Rational or Irrational?

To understand why, let’s delve into the definitions of rational and irrational numbers.

• Rational numbers are those that can be expressed as a fraction of two integers, where the denominator is not zero. In other words, they can be written in the form a/b, where a and b are integers and b is not equal to zero.

• An irrational number is a real number that cannot be expressed as a simple fraction, and its decimal representation goes on infinitely without repeating.

Since 72 is not a perfect square, its square root cannot be expressed as a fraction of two integers. Additionally, the decimal representation of √72 is non-repeating and non-terminating. Therefore, √72 is classified as an irrational number.

The square root of 72 (√72) is irrational because it cannot be expressed as a simple fraction. In decimal form, its digits continue infinitely without repeating any pattern. This is because 72 is not a perfect square, meaning it is not the square of an integer. Therefore, the square root of 72 (√72) is classified as an irrational number, which is a fundamental concept in mathematics with implications in various calculations and theoretical contexts.

Methods to Find Value of Root 72

Estimation Method:

• Start by finding the nearest perfect squares around 72. In this case, the nearest perfect squares are √64 = 8 and √81 = 9.
• Since 72 is closer to 81, start with an initial estimate of √72 ≈ 8.5.
• Refine the estimate iteratively using trial and error until you reach a satisfactory approximation.

Long Division Method:

• Use the long division method to approximate the square root of 72 manually.
• Start with an initial guess, such as 8, and proceed with division and adjustment until you achieve the desired accuracy.
• This method involves a series of steps of trial and error to converge on an approximation of √72.

Newton’s Method:

• Apply Newton’s method, an iterative algorithm, to approximate the square root of 72.
• This method involves refining an initial guess through successive iterations until reaching a sufficiently accurate approximation.
• Newton’s method is more efficient but requires a deeper understanding of calculus and iterative algorithms.

Using a Calculator or Software:

• Utilize a scientific calculator or mathematical software to directly calculate the square root of 72.
• Input 72 into the calculator or software, and the result will provide the accurate value of √72.

Square Root of 72 by Long Division Method

1. Digit Pairing: Pair off the digits of the number 72, starting from the right, and place a horizontal bar to indicate pairing.
2. Initial Estimation: Find a number that, when squared, gives a value less than or equal to 72. In this case, 8 fits as 8×8 = 64. Therefore, the initial quotient and divisor are both 8.
3. Iteration: Bring down two zeros to make the dividend 7200. Multiply the current quotient by 2 to get the next divisor, 16.
4. Adjustment: Determine the next digit of the quotient by finding the largest digit that, when appended to the current divisor, gives a product less than or equal to 7200. Continue this process iteratively.
5. Repeat: Repeat the steps of bringing down zeros, multiplying the current quotient by 2 to obtain the new divisor, and finding the next digit of the quotient until the desired level of precision is achieved.

By following these steps iteratively, the square root of 72 can be approximated using the long division method.

72 is Perfect Square root or Not

A perfect square is a number that can be expressed as the square of an integer. In other words, it is the product of an integer multiplied by itself. For example, 25 is a perfect square because it equals 5×5.

However, 72 cannot be expressed as the product of an integer multiplied by itself. Therefore, it is not a perfect square.

FAQ’s

How do I approximate the square root of 72 without a calculator?

You can use methods like estimation, long division, or iterative algorithms like Newton’s method to approximate the square root of 72 manually.

What are the practical applications of the square root of 72?

The square root of 72 has applications in various fields such as mathematics, physics, engineering, and finance, where precise calculations are required.

Are there any interesting mathematical facts related to the number 72?

Yes, 72 has several mathematical properties, such as being divisible by 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 itself. It is also the sum of four consecutive prime numbers (13 + 17 + 19 + 23 = 72).