Square Root of 1681

The square root is the inverse of the square of a number. 1681 is a perfect square. The square root of 1681 can be expressed in both radical and exponential form. In radical form, it is expressed as √1681, whereas in exponential form, it is expressed as (1681)^(1/2). The square root of 1681 is 41, which is a rational number because it can be expressed as a fraction where both the numerator and the denominator are integers.

The prime factorization method is commonly used for perfect square numbers. However, for educational purposes, the long division method and approximation method can also be used. Let us now learn these methods:

• Prime factorization method
• Long division method
• Approximation method

The prime factorization of a number involves breaking it down into its prime factors. Let's see how 1681 is factored:

Step 1: Finding the prime factors of 1681. Breaking it down, we find 1681 = 41 × 41.

Step 2: Since 1681 is a perfect square, the digits can be grouped into pairs, and each pair repeated.

Therefore, the square root of 1681 using prime factorization is 41.

The long division method is particularly useful for finding the square roots of non-perfect square numbers, but it is shown here for thoroughness.

Step 1: Pair the digits of 1681 from right to left, giving us two pairs: 16 and 81.

Step 2: Find a number whose square is less than or equal to 16. That number is 4, as 4 x 4 = 16. Subtract 16 from 16, leaving 0.

Step 3: Bring down the next pair, 81, to make it 081.

Step 4: Double the current quotient, 4, to make it 8. We now need to find a digit, say n, such that 8n × n ≤ 81. The number 1 works, as 81 x 1 = 81.

Step 5: Subtract 81 from 81, leaving 0.

The quotient is 41, which is the square root of 1681.

The approximation method is not necessary for perfect squares but can demonstrate the closeness of the square root for numbers similar to 1681.

Step 1: Identify the closest perfect squares around 1681. Here, 1600 (40^2) and 1764 (42^2) are the nearest perfect squares. √1681 lies between 40 and 42.

Step 2: Check the number halfway between these squares, which is 41.

Since 41 x 41 = 1681, we confirm that the square root of 1681 is 41.

• Square root: A square root is the inverse operation of squaring a number. For example, if 5^2 = 25, then √25 = 5.
   
• Rational number: A rational number is a number that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.
   
• Integer: An integer is a whole number that can be positive, negative, or zero. Examples include -3, 0, 4.
   
• Exponential form: A way of expressing numbers using a base raised to an exponent. For example, the square of 4 can be expressed as 4^2.