Square root of 324

The square root is a number that, when multiplied by itself, results in the original number whose square root is to be found. Know that the square root of 324 is ±18.

We will see here more about the square root of 324. As defined, the square root is just the opposite (inverse) of squaring a number, so, squaring 18 will result in 324. The positive value, 18 is the solution of the equation x² = 324.

It contains both positive and a negative root, where the positive root is called the principal square root. The square root of 324 is expressed as √324 in radical form. In exponential form, it is written as (324)^1/2  .

The simplest radical form of √324 is 18. We will see how to obtain the simplest radical form further in this article.
We now came to a point where we can say that:

• 324 is a perfect square.

• √320 is a rational number.
   

Let us now find how we got this value of 18 as a square root of 324.

We will use these methods below to find.

•  Prime factorization method

•  Long division method

• Approximation/Estimation method
   

The prime factorization of 324 involves breaking down a number into its factors.

Factorize 324 by prime numbers, and continue to divide the quotients until they can’t be separated anymore. 

• Find the prime factors of 324.

• After factoring 324, make pairs out of the factors to get the square root.

• If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.

Prime factorization of 324 = 2×2×3×3×3×3

For 324, three pairs of factors 2 and 3 are obtained.

So, it can be expressed as √324 = √(2×2×3×3×3×3) = 2×3×3 = 18.

Long Division method is used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder for non-perfect squares.

To make it simple it is operated on divide, multiply, subtract, bring down and do-again.

To calculate the square root of 324:

Step 1: On the number 324.00, draw a horizontal bar above the pair of digits from right to left.

Step 2 :Find the greatest number whose square is less than or equal to 3. Here, it is 1, Because 1²=1 < 3.

Step 3 : Now divide 3 by 1 such that we get 1 as a quotient and then multiply the divisor with the quotient, we get 1.

Step 4: Subtract 1 from 3. Bring down 2 and 4 and place it beside the difference 2.

Step 5: Add 1 to the same divisor, 1. We get 2.

Step 6: Now choose a number such that when placed at the end of 2, a 2-digit number will be formed. Multiply that particular number by the resultant number to get a number less than or equal to 224. Here, that number is 8. 

28×8=224. In quotient’s place, we also place that 8.

Step 7 : The quotient obtained is the square root. In this case, it is 18

We know that the sum of the first n odd numbers is n². We will use this fact to find square roots through the repeated subtraction method. Furthermore, we just have to subtract consecutive odd numbers from the given number, starting from 1. The square root of the given number will be the count of the number of steps required to obtain 0.

Here are the steps:

Step 1: take the number 324 and then subtract the first odd number from it. Here, in this case, it is 324-1=323.

Step 2: we have to subtract the next odd number from the obtained number until it comes  zero as a result. Now take the obtained number (from Step 1), i.e., 323, and again subtract the next odd number after 1, which is 3, → 323-3=320. Like this, we have to  proceed further.

Step 3: Now we have to count the number of subtraction steps it takes to yield 0 finally. Here, in this case, it takes 18 steps .

So, the square root is equal to the count, i.e., the square root of 324 is ±18.

• Negative Square root - The negative square root of 324 is √-324.

• Trigonometry - This branch of mathematics deals with the functions of angles.

• Exponential form - An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors, or a way to write a number that is multiplied by itself more than one time.  Ex: 5× 5 × 5 × 5 = 625 Or, 5 ⁴ = 625, where 5 is the base, 4 is the exponent 

• Prime Factorization - Expressing the given expression as a product of its factors Ex: 52=2 × 2 × 13 

• Prime Numbers - Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....

• Rational numbers - The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers

• Irrational numbers - Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. 

• Perfect square numbers - Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25.

• Non-Perfect square numbers -  Those numbers whose square roots comprise decimal places. Ex : 3, 10, 29