Square Root of 68

The square root is the inverse of the square of a number. 68 is not a perfect square. The square root of 68 is expressed in both radical and exponential form.

In radical form, it is expressed as √68, whereas in exponential form it is expressed as (68)^1/2. √68 ≈ 8.24621, 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:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 68 is broken down into its prime factors.

Step 1: Finding the prime factors of 68 Breaking it down, we get 2 x 2 x 17: 2² x 17¹

Step 2: Now we found out the prime factors of 68. The second step is to make pairs of those prime factors. Since 68 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 68 using prime factorization directly to find a simple square root is not feasible.

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 68, we consider it as a whole.

Step 2: Now we need to find n whose square is less than or equal to 68. We can say n as ‘8’ because 8 x 8 = 64, which is less than 68. Now the quotient is 8 after subtracting 64 from 68, the remainder is 4.

Step 3: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 400.

Step 4: Double the quotient (8) to get the new divisor's starting digits, which gives us 16_. We need to find a digit d such that 16d x d ≤ 400.

Step 5: By trial, 162 x 2 = 324, which is less than 400.

Step 6: Subtract 324 from 400 to get 76, and bring down two zeros to make it 7600.

Step 7: Repeat the process to find the next digit, continuing until the desired precision is reached. So the square root of √68 ≈ 8.246

The approximation method is another method for finding 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 68 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √68. The smallest perfect square less than 68 is 64, and the largest perfect square more than 68 is 81. √68 falls somewhere between 8 and 9.

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

Using the formula: (68 - 64) / (81 - 64) = 4/17 ≈ 0.235 Using the formula, we identified the decimal point of our square root.

The next step is adding the value we got initially to the smaller perfect square root number which is 8 + 0.235 = 8.235, so the approximate square root of 68 is 8.235.

• Square root: A square root is the inverse operation of squaring a number. For example, 4² = 16, and the inverse operation is the square root: √16 = 4.

• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction, i.e., in the form of p/q, where p and q are integers and q ≠ 0.

• Radical: A radical is an expression that uses the root symbol (√) to indicate the root of a number.

• Integer: An integer is a whole number that can be positive, negative, or zero, such as -5, 0, and 3.

• Decimal: A decimal is a number that includes a whole number and a fractional part separated by a decimal point, such as 7.86 or 9.42.