Square Root of 81

The square root of 81 is ±9, where 9 is the positive solution of the equation x² = 81. Finding the square root is just the inverse of squaring a number and hence, squaring 9 will result in 81. The square root of 81 is written as √81 in radical form, where the ‘√’  sign is called the “radical” sign. In exponential form, it is written as (81)^1/2 

We can find the square root of 81 through various methods. They are:

• Prime factorization method

• Long division method

• Repeated subtraction method
   

The prime factorization of 81 can be found by dividing the number by prime numbers and continuing to divide the quotients until they can’t be separated anymore.

Steps for Prime Factorization of 81:

Step 1:  Find the prime factors of 81.

Step 2: After factorizing 81, make pairs out of the factors to get the square root.

Step 3: If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.

 Step 4:If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.

This 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 too sometimes.

Follow the steps to calculate the square root of 81:

  Step 1: Write the number 81 and draw a bar above the pair of digits from right to left.

 Step 2: Now, find the greatest number whose square is less than or equal to 81. Here, it is 9 because 9²=81 

Step 3: now divide 81 by 9 (the number we got from Step 2) such that we get 9 as a quotient, and we get a remainder 0. 
 
Step 4: The quotient obtained is the square root of 81. In this case, it is 9.

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 81 and then subtract the first odd number from it. Here, in this case, it is 81-1=80

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., 80, and again subtract the next odd number after 1, which is 3, → 80-3=77. 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 9 steps 

So, the square root is equal to the count, i.e., the square root of 81 is ±9.

• Exponential form:  An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent. Ex: 2 × 2 × 2 × 2 = 16 Or, 2⁴ = 16, where 2 is the base, 4 is the exponent 

• Prime Factorization:  Expressing the given expression as a product of its factors. Ex: 48=2 × 2 × 2 × 2 × 3

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

• Rational numbers and Irrational 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. 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 and non-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 are those numbers whose square roots comprise decimal places. Ex :3, 8, 2