Square Root of 84
84 is the sum of twin primes (41 + 43). It is an even composite number that has 2, 3, and 7 as its prime factors. In this mini lesson, let us learn about the square root of 84, find out whether the square root of 84 is rational or irrational, and see how to find the square root of 84 by long division method.
 Square Root of 84: √84 = 9.165
 Square of 84: 84^{2} = 7056
1.  What Is the Square Root of 84? 
2.  Is Square Root of 84 Rational or Irrational? 
3.  How to Find the Square Root of 84? 
4.  FAQs on Square Root of 84 
What Is the Square Root of 84?
Finding the square root of a number, say n, is finding out what number, say a, multiplied by itself equals the number n. a × a = n ⇒ a^{2} = n. Thus a = √n. √84 = √(a × a )
 84 = 9.165 × 9.165 and 9.165 × 9.165
 √84 = ± 9.165
 We know that 84 = 2 × 2 × 3 × 7
 In the simplest radical form √84 = √(2 × 2 × 3 × 7) = 2√21
Is Square Root of 84 Rational or Irrational?
√84 = 9.1615138991 We cannot write this as a rational number of the form p/q. This is a nonterminating decimal. Thus the square root of 84 is irrational.
How to Find the Square Root of 84?
The square root of 84 or any number can be calculated in many ways. Two of them are the approximation method and the long division method.
Square Root of 84 by Approximation Method
 Take two perfect square numbers, one of which is just smaller than 84 and the other is just greater than 84. √81 < √84 < √100
 9 < √ 84 < 10
 Using the average method, divide 84 by 9 or 10.
 Let us divide by 10⇒ 84 ÷ 10 = 8.4
 Find the average of 8.4 and 10.
 (8.4 + 10) / 2 = 18.4 ÷ 2 = 9.2
 √84 ≈ 9.2
Square Root of 84 by the Long Division Method
The long division method helps us find a more accurate value of square root of any number. Let's see how to find the square root of by the long division method.
 Write 84 as 84. 00 00 00. Find a number × number that gives the product 84 or less than that.
 We determine 9 × 9 = 81. Subtract this from 84 and get the remainder as 3. Get down the pair of zeros down. 3 00 is our new dividend.
 Double the quotient. We obtain 18. Have 18x × x is less than or equal to 3 00.
 We determine that 181 × 1 = 181. Subtract this from 300 and obtain the remainder as 1 19. Bring down the next pair of zeros. 1 19 00 is the new dividend.
 Double the quotient 9.1⇒ 182 and let us have 182x × x as our new divisor.
 We determine 1826 × 6 = 1 09 56 as our product. Subtract this from 1 19 00. We obtain 9 44 as the remainder. Bring down the next pair of zeros.
 Repeat the process until we obtain the quotient approximated to 3 decimal places.
 √84 = ± 9.165
Explore square roots using illustrations and interactive examples
Tips and Tricks
The square root of any number can be assumed to be between the square root of the two nearest perfect squares of that number. For example, the square root of 108 lies between the square root of 100 and 121. Therefore, 10 < √108 < 11. Use the average method then, to evaluate the approximate value of √108.
The square root of 84 is evaluated using the division method and rounded off to the nearest hundredth. √84 = 9.165. We round it off to the nearest hundredth as 9.17.
Important Notes
 The square root of 84 is 9.165.
 The simplified form of radical form is 2√21
 √84 is an irrational number.
Square Root of 84 Solved Examples

Example 1: Charlie has made 84 cookies. If he has to arrange them on the tray as many cookies as the number of rows, how can he arrange them? How many cookies will be left out of this arrangement?
Solution: Number of cookies per row × number of rows = Total cookies
Let cookies per row = number of rows = n
n × n = 84
n^{2} = √84
n = 9.1 (approximated to the nearest tenth)
He can arrange 81 cookies in 9 rows and 3 cookies will be left out of the arrangement.

Example 2: Sam is playing with his blocks. He has built 7 blocks in a row and extended the shape in 12 columns.
a) How many blocks does he need to remove to make the rectangle to a square?
b) How many more blocks does he need to make this rectangle to a square?
Solution:
7 blocks in a row × 12 columns = Total number of blocks
7 × 12 = 84 blocks
a) He has to arrange them as a square base. n × n = 84
Since 84 is not a perfect square, let us make it a perfect square.
n × n = 81. We subtract 3 from 84 to make it a perfect square. 84  3 = 81
Thus he has to remove 3 blocks to build it as a square.
b) He has to arrange them as a square base. n × n = 84
Since 84 is not a perfect square, let us make it a perfect square.
n × n = 100. We add 16 to 84 to make it a perfect square. 84 + 16 = 100
Thus he has to add 16 blocks more to build it as a square.
FAQs on Square Root of 84
What is the square root of 84?
The square root of 84 is 9.165
What is the square root of 84 simplified?
2√21 is the simplest form of the square root of 84 in its radical form.
Is the square root of 84 real?
The square root of 84 is real.
Is the square root of 84 a rational number?
√84 is an irrational number, because the value of √84 is a nonteminating decimal.
How to find the square root of 84 to the nearest hundredth?
√84 rounded to the nearest hundredth as 9.17. we can use long division or approximation method to find it.
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