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In Mathematics, the square root of 576 is a value which when multiplied by itself gives the number 576. For example, 24 × 24 = 576. Accordingly, the square root of 576 is 24. Therefore, we can say the value of the square root of 576 is equivalent to 24. It is represented as √576 = 24 in radical form. There are two procedures to find the square root of 576. The two procedures to determine the square root value of 576 are Prime Factorization Method and Long Division Method. In this article, we will discuss both the procedures to find the square root that will help the students to calculate the square root of any numbers speedily and easily.

On the basis of the above equation, we can define that the square root of natural number 576 is 24. Digit 24 is a value which when multiplied by itself gives the result 576.

Hence, 24 × 24 = 576.

The square root of a number is another number which obtains the initial number when it is multiplied by itself. For example, 5× 5 = 25. Hence, the value of the square root of 25 is 5.

In Mathematics, the symbol used to denote square root is ‘ √ ’. This symbol of square roots is also known as radical. The number written inside the square root symbol is known as radicand.

The two methods to calculate the square root of a given natural number are:

Prime Factorization Method

Long Division Method

Prime Factorization Method

In Mathematics, the prime factorization method is a method to calculate the prime factors of a given number. The prime factorization method is simple to use as we have read about the prime factors in our previous classes. The prime factorization method can only be applied if the number given is a perfect square. A number calculated by squaring a number is known as a perfect square. For example, 16, 25, 576 etc are perfect squares.

As we know, 576 is a perfect square

\[576 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3\]

If we take the square root of both the left and right hand sides, we get

\[\sqrt{576} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3\]

We can see 3 pairs of 2 and 1 pair of 3 in the above given prime factors of 576.

\[\sqrt{576} = 2 \times 2 \times 2 \times 3\]

\[\sqrt{576} = 24\]

Hence, the square root of 576 is 24.

Long Division Method

Here, we will discuss the steps to find the square roots of 576 using the long division method.

Arrange 576 in pairs of two digits from right to left side.

\[\overline{05}\] \[\overline{76}\]

We will start with set 1, the largest perfect square less than or equal to 5 is 4 and the square root of 4 is 2. Place the digit 4 at the bottom and 2 at the top as shown in the image below.

2

\[\overline{05}\] \[\overline{76}\]

4

Subtract the digit 4 from the digit 5 and write the difference below. Now, examine the second set of numbers and move the second set of numbers.

2

\[\overline{05}\] \[\overline{76}\]

4

\[\overline{1}\] \[\overline{76}\]

Find the square of digit 2 which is marked in green at the top. Then use the digit 4 and the number placed at bottom i.e.176 to make the following equation.

4? × ? ≤ 176

By using the trial and error method, we can find the largest number and fill the blank of the above equation. Replace the question marks in the equation given above with 4 to get:

44 × 4 = 176.

Now, place the digit 4 on top, and 176 at the bottom:

2 4

\[\overline{05}\] \[\overline{76}\]

4

\[\overline{1}\] \[\overline{76}\]

\[\overline{1}\] \[\overline{76}\]

The difference between the bottom two numbers is zero. Hence, we found the answer which is marked in green at the top. The square root of 576 is 24.

1. Calculate the square root of 256 by using the prime factorization method.

Solution:

Split the number 256 by using the prime factorization method.

(image will be uploaded soon)

256 can also be written as = 2 × 2 ×2 ×2 × 2 × 2 × 2 × 2.

\[\sqrt{256} = \overline{2 \times 2}, \overline{2 \times 2}, \overline{2 \times 2}, \overline{2 \times 2}\]

\[\sqrt{256} = 2 \times 2 \times 2 \times 2\]

\[\sqrt{256} = 16\]

2. Find the square root of 900 through the prime factorization method.

Solution:

(image will be uploaded soon)

900 can be expressed as = 2 × 2 × 3 × 3 × 5 × 5

\[900 = 2^{2} \times 3^{2} \times 5^{2}\]

\[\sqrt{900} = \sqrt{2^{2} \times 3^{2} \times 5^{2}}\]

= \[2 \times 3 \times 5\]

= 30

3. Find the square root of 97344 using the long division method.

Solution:

(image will be uploaded soon)

Hence, the square root of 97344 is 312

\[\sqrt{97354} = 312\]

1. Which number will be at the unit place, when 5278 is squared.

8

7

6

4

2. Calculate the number of zeros in the square of 400.

2

3

4

6

FAQ (Frequently Asked Questions)

1. What are the Different Properties of a Square Root?

Ans: The different properties of a square root of a number are given below:

If the unit digit of numbers is 2,3,7 or 8 then it does not have a perfect square root.

For example, 132, 423, etc does not have a perfect square root as the unit digit of these numbers ends with 2,3,7 or 8.

The square root of an even number is always even and the square root of odd numbers is always odd. For example √144 = 12 ( both are even numbers).

The total of first n odd numbers is always n².

For example, 1 + 3 + 5 = 9 = 3² as there are three odd numbers so it will be 3² .

Negative numbers do not have a square root in the system of rational numbers. For example, √-900 is not a natural number. It is a complex number.

If a number has a square root in the set of natural numbers, then its unit digit must be 0,1,4,5, or 9. For example, the unit digit of √1024 is 2 as the unit digit of 1024 is 4 and its square root is 2.

The number does not have a square root if it ends with an odd number of zeroes. If the number ends with an even number of zeroes, then it has a square root in which the number of zeros at the last is half the number of zeroes in the number. For example, 2000 does not have a square root as the number of zeros at its end is odd and i.e is 3. The number 900 has a perfect square root as the number of zeros at the end is even and i.e is 2. Therefore, the square root of 900 is 30 and it has only 1 zero. √900= 30.

2. What are the Different Applications of Square Root in Real-Life?

Ans: The applications of the square root are applied in diverse fields. The most basic application of a square root is to calculate the sides of a square when the area of a square is known. Some other applications of square roots are as follows:

To determine the length of the diagonal of a square or a rectangle.

To solve quadratic equations.

Pythagorean theorem is used to calculate the third sides of a right-angle triangle and when any 2 sides of a right-angle triangle are known.

The square root is used to calculate the square root of standard deviation from the variance of a given data set.

Architects use square roots to solve the quadratic equation as square roots are usually used in their work.

The square roots are also used by carpenters because they usually use the pythagorean theorem to determine the materials for their jobs.