What is the Square Root of 144?

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 = \[\sqrt{b}\]. 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 ‘\[\sqrt{}\]’.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

\[\sqrt{144}\] = \[\sqrt{2\times2\times2\times2\times3\times3}\]

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.

Long Division Method

The square root of a number can also be calculated using the long division method. The long division method of finding the square root is useful and the square root of any number can be found out speedily. This method is primarily used to find the square roots of imperfect squares and large numbers which cannot be calculated using the prime factorization method. Below are some of the steps which show the way to calculate square root of 144 using the long division method.

Omit the first digit i.e.1 and leave the remaining two digits i.e. 44.

We know the square root of digit 1 is always 1. Hence, considering 1 as quotient, dividend and divisor, we get the remainder equals to 0.

Now, we will carry forward the other two numbers i..e. 44. The number 44 will be considered as dividend and we will further add 1 to the divisor to get our next divisor i.e.1 + 1= 2.

As the last digit is 4, then either the square of 2 or 4 can have the last digit 4.

In this step, we will fix 2 with 2 and multiply the divisor 22 by 2 such as 22 × 2 to get the dividend 44.

Hence, we get quotient as 12 and this will be our final answer.

Solved Examples

Simplify the following square roots

1. Find the Square Root of 36 By the Prime Factorization Method.

Solution:

1. Mod the number 36 into prime factorization using the method of synthetic division.

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\[\sqrt{36}\] = \[\sqrt{(2 \times 2 \times 3 \times 3)}\]

\[\sqrt{36}\] = (2 × 3)

\[\sqrt{36}\] = 6

2. Find the Square Root of 324 Using the Prime Factorization Method.

Solution:

324 can be written as 2 × 162

324 can also be written as 2 × 2 × 81

324 can also be written as 2 ×2 ×3 × 27

324 can also be written as 2 ×2 ×3 ×3 ×9

324 can also be written as 2 ×2 ×3 × 3 × 3 ×3

Therefore, \[\sqrt{324}\] = \[\bar{2\ast2}\] \[\ast\] \[\bar{3\ast3}\] \[\ast\] \[\bar{3\ast3}\]

= \[\sqrt{324}\] = 2 × 3 × 3

= \[\sqrt{324}\] = 18

3. Find the Square Root of 12544 Using the Long Division Method.

Solution:

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Hence, the square root of 12544 is 112

\[\sqrt{12544}\] = 112

Quiz Time

1. What will be the Value of 9² -1

81

80

79

None of the above

2. What will be the Square Root of 42?

1764

1664

1564

1504

FAQ (Frequently Asked Questions)

1. What is the Meaning of Prime Factorization?

Prime Factorization is a method of determining prime factors of a given number i.e the two prime numbers which when multiplied together give back an original number. For example, the prime factors of a number 126 will be 2, 3 ,and 7 as 2 × 3 × 7 = 126 and 2,3,and 7 are prime numbers.

Any natural or whole numbers (excluding 0) when multiplied by any prime numbers give composite numbers. Hence, we can say that prime factorization is generally implemented on composite numbers. The Prime Factorization method is also used in calculating HCF and LCM of any given set of numbers.

If two numbers are given, then the HCF (highest common factor ) will be the largest factor comprising both the numbers whereas the LCM (Least Common Multiple) will be the smallest common multiple existing in both the numbers.

2. What are the Different Steps to Calculate the Factors of 144?

The different steps to calculate the factors of 144 are given below:

Write down the number 144.

Find any two numbers which when multiplied together obtains the original number 144. For example, 2 and 72 are two numbers which when multiplied together obtains the result 144.

As we know, that 2 is the prime number which has two prime factors i.e, 2 and 1 and it cannot be factored further.

2 =2 ×1

On the other hand, 72 is a composite number that can easily be factored further. Hence, 72 can be factorized as 2 ×2× 2 × 3 × 3 × 1

Therefore, the factors of 144 can be written as 2 ×2 × 2 × 2 × 3 × 3 × 1.

At the end, mention all the unique numbers that are obtained as factors.