How to Calculate the Square Root of 144 with Examples and Tricks
In Mathematics, the square root of 144 is a value which when multiplied by itself gives the result 144. For example, 12 ×12= 144. Hence, we can see the square root value of 144 is 12. It is written in the radical form as
\[\sqrt{144}=12\].
There are two methods to determine the square root value of 144. The two methods to determine the square root value of 144 are Prime Factorization Method and Long Division Method. In this article, we will learn both the methods that will help the students to find the square root of any numbers in a simpler and fastest way.
According to the equation given above, we can state that the square root of natural number 144 is 12. It is a value which when multiplied by itself gives the number 144.
Hence, 12 ×12 = 144.
What is the Meaning of Square Root?
The square root of any natural number is a value that is represented in the form of \[X=b\sqrt{b}\]
It implies that x is the square root of b , where b is any natural number. We can also write it as x² = b. Hence, it is concluded that the square root of any number is equal to a number which when multiplied by itself obtains back the original numbers. For Example, 5 × 5 = 25 and it can be said that the square root of a number 25 is 5.
The symbol used to represent the square root is ‘√’
The symbol of square roots is also known as radical. The number inside the square root is known as radicand.
Square Root Methods
The two methods to find the square root of a given number are:
Prime Factorization Method
Long Division Method
Prime Factorization Method
In the Prime factorization method, we generally determine the prime factors of a given number. The prime factorization method can easily be used as we have studied about prime factors in our previous classes. This method can only be used if the number given is a perfect square. A number calculated by squaring a number is considered as a perfect square. A perfect square is an integer whose square root is always an integer . For example, 9, 36, 144 etc are perfect squares.
As we know,144 is a perfect square.
Therefore, the prime factors of 144 = 2 × 2 × 2 × 2 × 3 × 3
If we take the square root of both the sides, we get
\[\sqrt{144}\] = \[\sqrt{2\times 2\times 2\times 2\times 3\times 3}\]
We can see 2 pairs of 2 and 1 pair of 3 in the above given prime factors of 144.
\[\sqrt{144}\] = 2 × 2 × 3
\[\sqrt{144}\]= 12
Hence, the square root of 144 is 12.
Method of Long Division
We may also use the long division approach to obtain the square root of any number. This procedure is quite useful and the quickest of all for locating the root. It can be used to find the root of imperfect squares and huge numbers, something that prime factorisation cannot do. The steps for using the long division approach are outlined below.
Take the first digit, 1, and leave the remaining two digits, 4, alone.
The square of 1 is now 1. As a result, if we use 1 as the divisor, quotient, and dividend, the residual is 0.
We'll now deduct the other two numbers as dividends and add 1 to the divisor to get our next divisor, which is 1+1 =2.
Because the last digit is 4, either the square of 2 or the number 8 can be used as the last digit.
As a result, we'll add 2 to 2 and multiply by 2 to get 44, as in 22 x 2.
As a result, the final quotient is 12, which is the answer.
The steps to find the root of 144 are listed below.
FAQs on Square Root of 144 Explained
1. What is the value of the square root of 144?
The square root of 144 is 12. Since 144 is a perfect square, its square root is a whole number. Mathematically, this is written as √144 = 12, because 12 × 12 = 144. It is also important to know that both +12 and -12 are square roots of 144, as (–12) × (–12) also equals 144.
2. How do you find the square root of 144 using the prime factorisation method?
To find the square root of 144 by prime factorisation, you can follow these steps as per the CBSE syllabus:
- First, determine the prime factors of 144: 144 = 2 × 2 × 2 × 2 × 3 × 3.
- Next, group the factors into identical pairs: (2 × 2) × (2 × 2) × (3 × 3).
- Take one factor from each pair: 2, 2, and 3.
- Finally, multiply these chosen factors together to get the square root: 2 × 2 × 3 = 12.
3. What are the steps to find the square root of 144 using the long division method?
The long division method is another standard technique taught in the NCERT syllabus for finding square roots. Here are the steps for √144:
- Group the digits in pairs from the right. For 144, the groups are '1' and '44'.
- Find the largest number whose square is less than or equal to the first group (1). This number is 1. Write '1' as the divisor and in the quotient.
- Subtract the square (1 × 1 = 1) from the first group, which leaves 0. Bring down the next pair of digits, '44'.
- Double the current quotient (1 × 2 = 2) to get the new divisor's tens digit. Find a digit 'x' for the units place such that 2x × x ≤ 44.
- The required digit is 2, as 22 × 2 = 44. Write '2' in the quotient.
- Subtract 44 from 44, leaving a remainder of 0. The final quotient, 12, is the square root of 144.
4. Is 144 considered a perfect square? Explain why.
Yes, 144 is a perfect square. A number is defined as a perfect square if its square root is an integer (a whole number). Since the square root of 144 is 12, which is an integer, 144 is classified as a perfect square. This is confirmed as 12 multiplied by itself (12²) equals 144.
5. Why does the square root of 144 have two answers, +12 and -12?
The square root of a positive number has two solutions, one positive and one negative, because squaring either type of number results in a positive product.
- A positive number multiplied by itself gives a positive result: (+12) × (+12) = 144.
- A negative number multiplied by itself also gives a positive result: (–12) × (–12) = 144.
Therefore, both +12 and –12 are correct square roots of 144. The positive value, 12, is referred to as the principal square root.
6. How does finding the square root of 144 differ from finding its cube root?
Finding a square root and a cube root are different mathematical operations that ask for different things:
- The square root of 144 (√144) asks for a number that, when multiplied by itself, equals 144. That number is 12.
- The cube root of 144 (³√144) asks for a number that, when multiplied by itself three times, equals 144. This number is not an integer; it is an irrational number approximately equal to 5.241.
7. Can you provide a real-world example where calculating the square root of 144 is necessary?
A common real-world application for square roots is in geometry and construction. For instance, if you have a square plot of land with a total area of 144 square feet and you want to determine the length of one of its sides to install a fence, you would need to calculate the square root of the area. Since the area of a square is calculated as side × side (or side²), the length of one side would be √144 = 12 feet.
8. Why is the prime factorisation method so effective for finding the square root of perfect squares like 144?
The prime factorisation method is effective because of the fundamental structure of a perfect square. By definition, a perfect square is an integer multiplied by itself. This implies that in its prime factorisation, every prime factor must appear an even number of times. For 144, the prime factors are 2, 2, 2, 2, 3, 3. The factor '2' appears four times and '3' appears twice, both even counts. Grouping these into two identical sets (e.g., (2×2×3) and (2×2×3)) effectively reverses the squaring process and reveals the integer root, which is 12.
