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: 84² = 7056

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 ⇒ a² = 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

 √84 =   9.1615138991 We cannot write this as a rational number of the form p/q. This is a non-terminating decimal. Thus the square root of 84 is irrational.

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

• Square root of 98
• Square root of 54
• Square root of 74
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• Square root of 120

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.

• The square root of 84 is 9.165.
• The simplified form of radical form is 2√21
• √84 is an irrational number.

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 non-teminating 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.