Square Root of 6984

The square root is the inverse operation of squaring a number. 6984 is not a perfect square. The square root of 6984 is expressed in both radical and exponential form. In radical form, it is expressed as √6984, whereas it is expressed as (6984)^(1/2) in exponential form. √6984 ≈ 83.553, which is an irrational number because it cannot be expressed as a ratio of two integers.

The prime factorization method is typically used for perfect square numbers. However, for non-perfect square numbers, the long division method and approximation method are often used. Let us now learn these methods:

• Prime factorization method
• Long division method
• Approximation method

Prime factorization involves expressing a number as a product of its prime factors. Let's see how 6984 can be broken down:

Step 1: Finding the prime factors of 6984 By dividing 6984, we get 2 x 2 x 2 x 3 x 3 x 97: 2^3 x 3^2 x 97

Step 2: Now that we have the prime factors of 6984, we attempt to make pairs of those prime factors. Since 6984 is not a perfect square, the digits cannot be perfectly grouped into pairs.

Therefore, calculating √6984 using prime factorization is not straightforward.

The long division method is particularly useful for non-perfect square numbers. In this method, we find the closest perfect square numbers for comparison. Let's learn how to find the square root using the long division method, step by step:

Step 1: Begin by grouping the numbers from right to left. For 6984, consider it as 69 and 84.

Step 2: Find n such that n^2 is less than or equal to 69. Here, n is 8, since 8^2 = 64. The quotient is 8, and the remainder is 5 after subtracting 64 from 69.

Step 3: Bring down 84 to make the new dividend 584. Add the old divisor 8 with itself to get 16 as the new divisor.

Step 4: Find n such that 16n × n is less than or equal to 584. Let's consider n as 3, so 16 x 3 x 3 = 144.

Step 5: Subtract 144 from 584, resulting in a remainder of 440. The quotient becomes 83.

Step 6: Add a decimal point and bring down two zeros to make the new dividend 44000.

Step 7: The new divisor is 166, and the value of n is 2 because 166 x 2 = 332.

Step 8: Subtract 332 from 440, resulting in a remainder of 108.

Step 9: The quotient is now 83.5.

Step 10: Continue this process until reaching the desired decimal places. Thus, √6984 ≈ 83.553.

The approximation method is a simple way to estimate square roots. Let's learn how to find the square root of 6984 using the approximation method.

Step 1: Find the closest perfect squares to √6984. The closest perfect squares are 6724 (82^2) and 6889 (83^2). √6984 falls between 82 and 83.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula (6984 - 6889) ÷ (6889 - 6724) = 0.553 Adding this to the smaller integer, we get 83 + 0.553 = 83.553. So, the square root of 6984 is approximately 83.553.

• Square root: A square root is a number that, when multiplied by itself, gives the original number. For example, √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be expressed as a ratio of two integers.
   
• Radical: The symbol √ used to denote the square root or nth root of a number.
   
• Decimal: A number that includes both a whole number and a fractional part, represented by digits following a decimal point.
   
• Long Division Method: A step-by-step method used to find the square root of a number that is not a perfect square.