How to Calculate the Square Root of 576 Easily
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.
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:
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.
\[\sqrt{576}=2\times 2\times 2\times 3\times 3\]
\[\sqrt{576}=24\]
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:
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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, 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.
FAQs on Square Root of 576 Explained
1. What is the square root of 576?
The square root of 576 is 24. This is because 24 is the number that, when multiplied by itself (24 × 24), equals 576. Since 24 is a whole number, 576 is known as a perfect square.
2. Why is 576 considered a perfect square?
A number is called a perfect square if its square root is an integer. For 576, its square root is 24, which is an integer. Because an integer (24) can be squared to get 576, the number 576 perfectly fits the definition of a perfect square.
3. How can you find the square root of 576 using the prime factorisation method?
To find the square root of 576 using the prime factorisation method, you follow these steps:
First, break down 576 into its prime factors: 576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3.
Next, group these prime factors into identical pairs: (2 × 2) × (2 × 2) × (2 × 2) × (3 × 3).
Take one factor from each pair: 2 × 2 × 2 × 3.
Finally, multiply these factors together to get the square root: 2 × 2 × 2 × 3 = 24.
4. What is the process to find the square root of 576 using the long division method?
The long division method for finding the square root of 576 involves these steps:
Group the digits in pairs from the right, which gives '5' and '76'.
Find the largest number whose square is less than or equal to the first group (5). This number is 2 (since 2²=4). This '2' becomes the first digit of our answer and also the divisor.
Subtract 4 from 5 to get a remainder of 1. Bring down the next pair '76' to make the new dividend 176.
Double the current quotient (2) to get a new divisor '4_'. Find a digit to fill the blank such that '4_ × _' is less than or equal to 176. The digit is 4, as 44 × 4 = 176.
The remainder is 0, and the final quotient is 24, which is the square root.
5. How can you tell if a number like 576 might be a perfect square just by looking at its last digit?
You can get a clue about whether a number is a perfect square by examining its unit digit. A number can only be a perfect square if its unit digit is 0, 1, 4, 5, 6, or 9. Since 576 ends in 6, it has the possibility of being a perfect square. Conversely, any number ending in 2, 3, 7, or 8 can never be a perfect square.
6. Does the square root of 576 have both a positive and a negative answer? Explain why.
Yes, mathematically, every positive number has two square roots: one positive and one negative. This is because both 24 × 24 = 576 and (-24) × (-24) = 576. However, when we use the radical symbol (√), it refers to the principal square root, which is the non-negative value. Therefore, √576 is conventionally written as 24.
7. What are some real-world applications where calculating the square root of a number like 576 is important?
Calculating square roots is important in many practical fields. For example:
Geometry and Land Measurement: If a square-shaped plot of land has an area of 576 square metres, you would find its side length by calculating √576, which is 24 metres.
Physics and Engineering: Square roots are used in formulas for calculating distance, speed, and in fields like electronics and signal processing.
Construction: Carpenters and architects use the Pythagorean theorem (a² + b² = c²), which involves square roots, to ensure right angles and calculate diagonal lengths.
