Square root of 89

The square root of 89 is ±9.43398113206. The positive value, 9.43398113206 is the solution of the equation x² = 89. As defined, the square root is just the inverse of squaring a number, so, squaring 9.43398113206 will result in 89.  The square root of 89 is expressed as √89 in radical form, where the ‘√’  sign is called “radical”  sign. In exponential form, it is written as (89)^1/2  .
 

We can find the square root of 89 through various methods. They are:

• Prime factorization method

• Long division method

• Approximation/Estimation method
   

The prime factorization of 89 involves breaking down a number into its factors. Divide 89 by prime numbers, and continue to divide the quotients until they can’t be separated anymore. 

• Find the prime factors of 89.

• After factorizing 89, 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.

So, Prime factorization of 89 = 89 × 1  

for 89, no pairs of factors are obtained, but a single 89 is there.

So, it can be expressed as  √89 = √(89 × 1) = √89

√89 is the simplest radical form of √89.

This is a method 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 89:

Step 1 : Write the number 89, and draw a horizontal 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 89. Here, it is 9, Because 9²=81 < 89.

Step 3 : Now divide 89 by 9 (the number we got from Step 2) such that we get 9 as quotient and then multiply the divisor with the quotient, we get 81. Subtract 81 from 89, we get 8. Add a decimal point after the quotient 9, and bring down two zeroes and place it beside 8 to make it 800.

Step 4: Add 9 to same divisor, 9. We get 18.

Step 5: Now choose a number such that when placed at the end of 18, a 3-digit number will be formed. Multiply that particular number by the resultant number to get a number less than 800. Here, that number is 4. 

184×4=736<800.

Step 6: Do 800-736=64. Again, bring down two zeroes and make 64 as 6400. Simultaneously add the unit’s place digit of 184, i.e., 4 with 184. We get here, 188. Apply Step 5 again and again until you reach 0. 

We will show two places of precision here, and so, we are left with the remainder, 751 (refer to the picture), after some iterations and keeping the division till here, at this point 

             
Step 7 : The quotient obtained is the square root. In this case, it is 9.43….

Approximation or estimation of square root is not the exact square root, but it is an estimate.Here, through this method, an approximate value of square root is found by guessing.

Follow the steps below:

Step 1: Find the nearest perfect square number to 89. Here, it is 81 and 100.

Step 2: We know that, √81=9 and √100=10. This implies that √89 lies between 9 and 10.

Step 3: Now we need to check √89 is closer to 9 or 9.5. Let us consider 9 and 9.5. Since (9)²=81 and (9.5)²=90.25. Thus, √89 lies between 9 and 9.5.

Step 4: Again considering precisely, find squares of (9.2)²=84.64 and (9.45)²= 89.30.

We can iterate the process and check between the squares of 9.3 and 9.44 and so on.

We observe that √89=9.43…

• Estimation Method:  Estimation refer to a reasonable guess of the actual value to make calculations realistic and precise. This method helps in estimating the square root of a number.

• 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: 3× 3 × 3 × 3 = 81 Or, 3⁴ = 81, where 3 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 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 :2, 8, 18