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Square Root of 576

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Last updated date: 23rd Apr 2024
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What is the Square Root of 576?

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.

The square root of \[576,\sqrt{576}\] is 24.

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.

 

What does Square Root Mean?

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.

 

Methods to Find Square Root

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

  1. Prime Factorization Method

  2. 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.

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

\[\sqrt{576}=24\]

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.

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

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

2. 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

3. 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           

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

4. 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

5. 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.

 

Solved Examples

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

\[\sqrt{97354}=312\]

 

576th Square Root

576 is divided by 24 to get the square root. It is written as 576 = 24 in radical form. There are two ways for calculating the value to root 576 that do not require the use of a calculator: prime factorisation and long division. We'll go over both ways here, which will enable students to discover the root of any number quickly and easily. Also, get a thorough understanding of square roots here.

The square root of \[576,\sqrt{576}\]is 24.

The square root of 576, as defined by the preceding equation, is a value (i.e. 24) that when multiplied by itself yields 576.

As a result, 24 × 24 = 576 (or 242 = 576).

 

The Square Root of 576 has been Simplified

We'll learn two distinct approaches to find the square root of 576 without using calculators.

 

Factorization in the First Place:

We shall write the prime factors of the given number using the prime factorisation method. Finding prime factors of numbers is something we've already learnt in earlier classes. However, this method can only be used to find square roots if the input number is a perfect square.

576 is clearly a perfect square, hence the prime elements are,

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

As a result, we get; if we take the square root on both sides.

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

There are three pairs of 2 and one pair of 3 as seen above.

Hence,

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

\[\sqrt{576}=24\]

 

Method of Long Division

The long division approach can also be used to obtain the square root of perfect squares. This is the quickest way to get the root of a number, and it's especially handy for imperfect squares and huge integers. The procedures for finding the value of 576 using the long division method are listed below.

  • For the division, take the first digit, 5, and leave the remaining two digits, 76.

  • The square of 2 is now 4. Using 2 as the divisor, 4 as the quotient, and 5 as the dividend, we get a remainder of 1.

  • We'll now deduct the other two numbers, 76 and 2, from the dividend and add 2 to the previous divisor to produce our next divisor, 2+2 =4.

  • Because the last digit is 6, either the square of 4 or the square of 6 can be used to get the last digit to be 6.

  • As a result, we'll combine 4 with 4 and multiply by 4 to get 176 (44 x 4).

  • As a result, the final quotient is 24, which is the correct answer.

The steps to find the root of 576 are listed below.


2 4

2

5 7 6

+2

4

44

1 7 6


1 7 6


X X X

FAQs on Square Root of 576

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

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

  1. 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.

  1. 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).

  2. 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 .

  1. 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.

  2. 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.

  3. 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?

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:

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

  2. To solve quadratic equations.

  3. 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.

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

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

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

3. What is the number one property of a Square root?

There are different properties of square roots. You can study about the methods to find square roots and their properties with the help of a free PDF available at Vedantu- Square Root of 576 - Methods to Find Square Root, Solved Examples and Important FAQs. The first property states that 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.

4. What is the second and third property of square roots?

The second property of square root is: 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 third one states that- 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. It can be well studied with the help of a free pdf available at Vedantu from Square Root of 576 - Methods to Find Square Root, Solved Examples and Important FAQs. 

5. What is the property for a negative number?

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. This can be well studied with the help of Square Root of 576 - Methods to Find Square Root, Solved Examples and Important FAQs. 

6. What numbers do not have a square root?

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.

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

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. It can be applied when- 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, and Architects use square roots to solve the quadratic equation as square roots are usually used in their work.