Square Root of 70

The square root is the inverse of the square of the number. 70 is not a perfect square. The square root of 70 is expressed in both radical and exponential form.

In the radical form, it is expressed as √70, whereas (70)^(1/2) is in exponential form. √70 ≈ 8.3666, 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:

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 70 is broken down into its prime factors.

Step 1: Finding the prime factors of 70 Breaking it down, we get 2 x 5 x 7.

Step 2: Now we found out the prime factors of 70. The second step is to make pairs of those prime factors. Since 70 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating √70 using prime factorization is impossible.

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 70, we need to group it as 70.

Step 2: Now we need to find n whose square is closest to 70. We can say n as 8 because 8 x 8 = 64, which is lesser than 70. Now the quotient is 8 after subtracting 70 - 64, the remainder is 6.

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 remainder. Now the new remainder is 600.

Step 4: Double the quotient and consider it as the new divisor. The new divisor will be 8 x 2 = 16. We need to find a digit n such that 16n x n ≤ 600. Let us consider n as 3. Then 163 x 3 = 489.

Step 5: Subtract 600 from 489, the difference is 111, and the quotient becomes 8.3

Step 6: Continue doing these steps until we get the desired decimal places. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √70 ≈ 8.3666.

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 70 using the approximation method.

Step 1: Now we have to find the closest perfect squares around 70. The smallest perfect square less than 70 is 64, and the largest perfect square greater than 70 is 81. √70 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, (70 - 64) ÷ (81 - 64) = 6 ÷ 17 ≈ 0.3529.

Adding this to the smaller perfect square root gives 8 + 0.3529 ≈ 8.3529.

• Square root: A square root is the number that gives the original number when multiplied by itself. For example, 4² = 16, and the inverse of the square is the square root, so √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, but the positive square root is more commonly used in real-world scenarios. This is known as the principal square root.

• Prime factorization: It is the process of expressing a number as the product of its prime factors. For example, the prime factorization of 70 is 2, 5, and 7.

• Long division method: A mathematical method used to find the square root of non-perfect squares by calculating each digit of the square root one at a time.