Square Root of 1602

The square root is the inverse operation of squaring a number. 1602 is not a perfect square. The square root of 1602 is expressed in both radical and exponential form. In radical form, it is expressed as √1602, whereas (1602)^(1/2) in exponential form. √1602 ≈ 40.0225, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

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

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Let us break down 1602 into its prime factors:

Step 1: Finding the prime factors of 1602 Breaking it down, we get 2 x 3 x 3 x 89: 2 x 3^2 x 89

Step 2: Now we found the prime factors of 1602. Since 1602 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating √1602 using prime factorization alone is not feasible.

The long division method is used for non-perfect square numbers. This method involves finding the closest perfect square number for the given number and using division to find the square root step by step.

Step 1: Begin by grouping the digits of 1602. Group it as 16 and 02.

Step 2: Find n whose square is less than or equal to 16. n is 4 since 4^2 = 16. The quotient is 4, and the remainder is 0.

Step 3: Bring down the next pair, 02, making the new dividend 02.

Step 4: Double the quotient (4) to get 8, which becomes part of the new divisor.

Step 5: Find the largest digit x such that 8x * x ≤ 02. Since 8 * 0 * 0 = 0 ≤ 02, x is 0.

Step 6: Subtract 0 from 02. The result is 02, and the quotient is 40.

Step 7: Add a decimal point and bring down two zeros. The new dividend is 200.

Step 8: Find the new divisor, which is 80. Find x such that 80x * x is less than or equal to 200. x is 2.

Step 9: Subtract 160 from 200, leaving a remainder of 40.

Step 10: Continue these steps until the desired decimal accuracy is achieved.

The square root of 1602 is approximately 40.0225.

The approximation method is a straightforward way to estimate square roots. Let's find the square root of 1602 using this method.

Step 1: Find the closest perfect squares to √1602.

The closest perfect squares are 1600 and 1681.

√1602 falls between 40 and 41.

Step 2: Use interpolation to find a more accurate value.

(Given number - smallest perfect square) / (larger perfect square - smaller perfect square).

(1602 - 1600) / (1681 - 1600) = 2 / 81 ≈ 0.0247

Adding this to the lower bound: 40 + 0.0247 ≈ 40.0247

Thus, the square root of 1602 is approximately 40.0247.

• Square root: A square root is the inverse of squaring a number. For example, 4^2 = 16, and the inverse is √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
   
• Perfect square: A number is a perfect square if it is the square of an integer. For example, 144 is a perfect square because 12^2 = 144.
   
• Decimal: A decimal number has a whole number and a fractional part separated by a decimal point. Examples include 7.86, 8.65, and 9.42.
   
• Interpolation: Interpolation is a method of estimating unknown values that fall between known values.