Square Root of 17.7

The square root is the inverse of the square of a number. 17.7 is not a perfect square. The square root of 17.7 is expressed in both radical and exponential form. In the radical form, it is expressed as √17.7, whereas (17.7)^(1/2) is its exponential form. √17.7 ≈ 4.20714, 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, the prime factorization method is not used for non-perfect square numbers, where long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 17.7, treat it as 1770 by considering it in the hundredths place.

Step 2: Now we need to find n whose square is less than or equal to 1. We can say n is '1' because 1 × 1 is less than or equal to 1. Now the quotient is 1, and after subtracting 1 - 1, the remainder is 0.

Step 3: Now let us bring down 77, making the new dividend 170. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 170. Let us consider n as 8; now 28 × 8 = 224, which exceeds 170. Try n as 6; 26 × 6 = 156, which fits.

Step 6: Subtract 156 from 170; the difference is 14, and the quotient is 16.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to bring down two zeroes, making the new dividend 1400.

Step 8: Now, the new divisor is 32 because 326 × 4 = 1304.

Step 9: Subtracting 1304 from 1400 results in 96.

Step 10: Now the quotient is 4.2.

Step 11: Continue doing these steps until you have two numbers after the decimal point or the remainder is zero.

So, the square root of √17.7 is approximately 4.21.

The approximation method is another method for finding the square roots; it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 17.7 using the approximation method.

Step 1: We have to find the closest perfect square of √17.7. The smallest perfect square less than 17.7 is 16, and the largest perfect square greater than 17.7 is 25. √17.7 falls somewhere between 4 and 5.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using this formula: (17.7 - 16) / (25 - 16) = 1.7 / 9 ≈ 0.1889

The next step is adding the value we got initially to the decimal number: 4 + 0.1889 ≈ 4.19, so the square root of 17.7 is approximately 4.19.

• Square root: A square root is the inverse of a square. For example, 4² = 16 and the inverse of the square is the square root, that is, √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is why it is also known as the principal square root.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.
   
• Long division method: A technique used to find more precise values of square roots for numbers that are not perfect squares by dividing and averaging.