Square Root of 867

The square root is the inverse operation of squaring a number. 867 is not a perfect square. The square root of 867 can be expressed in both radical and exponential forms. In radical form, it is expressed as √867, whereas in exponential form, it is (867)^(1/2). The approximate value of √867 is 29.445, which is an irrational number because it cannot be expressed as a fraction of two integers.

For perfect square numbers, the prime factorization method can be used. However, for non-perfect square numbers like 867, the long division and approximation methods are more suitable. Let us explore these methods: Prime factorization method Long division method Approximation method

The prime factorization of a number involves expressing it as a product of prime numbers. Let's examine the prime factorization of 867:

Step 1: Finding the prime factors of 867

Breaking it down, 867 = 3 × 17 × 17.

Step 2: Since 867 is not a perfect square, its digits cannot be grouped into pairs of equal factors. Thus, calculating √867 using prime factorization is not straightforward, but it helps in verifying results obtained by other methods.

The long division method is useful for non-perfect square numbers. Here's how to find the square root of 867 step by step:

Step 1: Begin by grouping the digits from right to left. For 867, we have 67 and 8.

Step 2: Find a number n whose square is less than or equal to 8. The number is 2 because 2 × 2 = 4. The quotient is 2, and the remainder is 8 - 4 = 4.

Step 3: Bring down the next group, 67, to make it 467. Add the previous divisor (2) to itself to get 4.

Step 4: Find a digit x such that 4x × x is less than or equal to 467. Let x be 9; then 49 × 9 = 441.

Step 5: Subtract 441 from 467; the remainder is 26.

Step 6: Add a decimal point, bring down 00 to make it 2600, and repeat the process.

Step 7: Continue this process to achieve the desired accuracy.

The quotient now is approximately 29.445.

The approximation method is a simpler way to estimate square roots. Here's how to approximate the square root of 867:

Step 1: Identify the perfect squares closest to 867. The perfect squares are 841 (29²) and 900 (30²), so √867 is between 29 and 30.

Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (867 - 841) / (900 - 841) = 26 / 59 ≈ 0.441

Step 3: Add this to the smaller root: 29 + 0.441 = 29.441

So the approximate square root of 867 is 29.441.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 × 4 = 16.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction. Its decimal goes on forever without repeating.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 9 is a perfect square because it is 3².
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors.
   
• Decimal: A decimal number is a number that consists of a whole number and a fractional part separated by a decimal point, such as 7.86 or 29.445.