Square Root of 4480

The square root is the inverse of the square of the number. 4480 is not a perfect square. The square root of 4480 is expressed in both radical and exponential form. In the radical form, it is expressed as √4480, whereas (4480)^(1/2) in the exponential form. √4480 ≈ 66.976, 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 typically used for perfect square numbers. However, for non-perfect square numbers like 4480, methods such as long division and approximation are more suitable. 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. Now let us look at how 4480 is broken down into its prime factors.

Step 1: Finding the prime factors of 4480 Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 5 × 7 × 7: 2^5 × 5^1 × 7^2

Step 2: Now we found out the prime factors of 4480. The second step is to make pairs of those prime factors. Since 4480 is not a perfect square, the digits of the number can’t be grouped in perfect pairs. Calculating √4480 using prime factorization requires approximation.

The long division method is particularly used for non-perfect square numbers. Let's 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 4480, we need to group it as 44 and 80.

Step 2: Now we need to find n whose square is less than or equal to 44. We use n as '6' because 6 × 6 = 36, which is less than 44. Now the quotient is 6, and after subtracting 36 from 44, the remainder is 8.

Step 3: Now let us bring down 80 to make the new dividend 880. Add the old divisor with the same number: 6 + 6 = 12, which will be our new divisor.

Step 4: The new divisor will be 12n. We need to find the value of n such that 12n × n ≤ 880. Let n be 7, then 127 × 7 = 889.

Step 5: Subtract 889 from 880; the difference is -9, and the quotient is 67.

Step 6: 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 900.

Step 7: Now we need to find the new divisor. Try n = 7, so 134 × 7 = 938.

Step 8: Subtracting 938 from 900, we get -38.

Step 9: Continue these steps until you have the desired precision. So the square root of √4480 ≈ 66.976.

The approximation method is another way of finding square roots; it is straightforward. Now, let us learn how to find the square root of 4480 using the approximation method.

Step 1: Find the closest perfect squares of √4480. The smallest perfect square less than 4480 is 4225 (65^2), and the largest perfect square more than 4480 is 4624 (68^2). Therefore, √4480 falls between 65 and 68.

Step 2: Use the interpolation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (4480 - 4225) / (4624 - 4225) = 0.639 Using the formula, we identified the decimal value to add to the initial estimate. Adding this gives us 65 + 1.639 = 66.639, so the square root of 4480 is approximately 66.976.

• Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the inverse operation is the square root: √16 = 4.

• Irrational number: An irrational number cannot be exactly expressed as a simple fraction (p/q), where q is not zero.

• Prime factorization: Breaking down a composite number into a product of its prime factors.

• Long division method: A method for finding the square root of non-perfect square numbers using a systematic division process.

• Decimal approximation: The process of estimating the value of a number to its nearest decimal form.