Square Root of 751

The square root is the inverse operation of squaring a number. 751 is not a perfect square. The square root of 751 is expressed in both radical and exponential form. In radical form, it is expressed as √751, whereas in exponential form it is expressed as (751)^(1/2). √751 ≈ 27.4014, which is an irrational number because it cannot be expressed as a fraction 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:

• 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 751 is broken down into its prime factors.

Step 1: Finding the prime factors of 751 Breaking it down, we find that 751 is itself a prime number.

Step 2: Since 751 is not a perfect square and is a prime number, we cannot group any digits into pairs.

Therefore, calculating the square root of 751 using the prime factorization method is not possible.

The long division method is particularly used for non-perfect square numbers. In this method, we find the closest perfect square numbers 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, group the numbers from right to left. For 751, we group it as 51 and 7.

Step 2: Find n whose square is less than or equal to 7. We can use n = 2 because 2^2 = 4 is less than 7. The quotient is 2, and after subtracting, the remainder is 3.

Step 3: Bring down 51, making the new dividend 351. Add the old divisor (2) to the same number to get 4, which becomes our new divisor.

Step 4: The new divisor is 4n. Find n such that 4n × n is less than or equal to 351. Let n = 7, then 47 × 7 = 329.

Step 5: Subtract 329 from 351; the difference is 22, and the quotient is 27.

Step 6: Since the dividend is less than the divisor, add a decimal point and two zeroes to the dividend, making it 2200.

Step 7: Find the new divisor, which is 27 (since 274 × 8 = 2192).

Step 8: Subtract 2192 from 2200, giving a remainder of 8.

Step 9: The quotient is now 27.4.

Step 10: Continue these steps until you achieve the desired precision.

So the square root of √751 ≈ 27.4014.

The approximation method is another way to find square roots. It is an easy method to find the square root of a given number. Let's learn how to find the square root of 751 using this method.

Step 1: Find the closest perfect squares to √751.

The smallest perfect square less than 751 is 729, and the largest perfect square above 751 is 784. √751 falls between 27 and 28.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (751 - 729) / (784 - 729) = 22 / 55 ≈ 0.4 Add this to the smaller square root: 27 + 0.4 = 27.4

Thus, the square root of 751 is approximately 27.4014.

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

• Irrational number: An irrational number is a number that cannot be expressed as a fraction of two integers (p/q, where q ≠ 0).

• Principal square root: The principal square root is the non-negative square root of a number.

• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

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