Finding the Square Root of 5776

The question asks us to find the square root of the number 5776. The square root of a number is a value that, when multiplied by itself, gives the original number. We are looking for a number 'x' such that \(x^2 = 5776\), or \(x = \sqrt{5776}\).

We can find the square root of 5776 using methods like prime factorization or the long division method for square roots. The long division method is often straightforward for finding the exact square root of a perfect square like 5776.

Step-by-Step Square Root Calculation of 5776 (Long Division Method)

Here's how to calculate the square root of 5776 using the long division method:

1. Group the digits in pairs, starting from the right. For 5776, we have the pairs 57 and 76.
2. Find the largest number whose square is less than or equal to the first pair (57). The numbers are: \(1^2=1\), \(2^2=4\), \(3^2=9\), \(4^2=16\), \(5^2=25\), \(6^2=36\), \(7^2=49\), \(8^2=64\). The largest square less than or equal to 57 is 49, which is \(7^2\).
3. Write 7 as the first digit of the square root. Subtract 49 from 57: \(57 - 49 = 8\).
4. Bring down the next pair of digits (76) next to the remainder 8. The new number is 876.
5. Double the current root (7), which gives 14. Write this down, followed by a blank space (representing a digit 'x'), so it looks like 14_.
6. Find a digit 'x' to place in the blank space such that when the new number (14x) is multiplied by 'x', the product is less than or equal to 876. The last digit of 876 is 6. The square of a number ending in 4 (\(4^2=16\)) or 6 (\(6^2=36\)) ends in 6. Let's try 6.
7. Place 6 in the blank space: 146. Multiply 146 by 6: \(146 \times 6\).
   • \(6 \times 6 = 36\) (write 6, carry 3)
   • \(6 \times 4 = 24 + 3 (\text{carry}) = 27\) (write 7, carry 2)
   • \(6 \times 1 = 6 + 2 (\text{carry}) = 8\) (write 8)
   The product is 876.
8. Subtract 876 from the current number 876: \(876 - 876 = 0\).
9. Since the remainder is 0 and there are no more pairs of digits to bring down, the process is complete.

The digits we found for the square root are 7 and 6. Therefore, the square root of 5776 is 76.

Let's verify the answer by squaring 76:

\(76 \times 76\)

We can calculate this as:

\[ \begin{array}{@{}c@{\,}c} & 76 \\ \times & 76 \\ \hline & 456 \quad (76 \times 6) \\ + & 5320 \quad (76 \times 70) \\ \hline & 5776 \\ \end{array} \]

The result is 5776, confirming that 76 is indeed the square root of 5776.

Based on the calculations, the square root of 5776 is 76.

Revision Table: Key Concepts for Square Roots

Additional Information: Why Long Division for Square Roots Works

The long division method for square roots is based on the algebraic expansion of a binomial squared: \((a+b)^2 = a^2 + 2ab + b^2\). In each step of the long division, we are essentially finding the next digit (b) of the root, given the digits found so far (a). The process iteratively finds the digits of the root from left to right, subtracting the square of the root found so far and using the remaining value to find the next digit.

For example, finding $\sqrt{5776}$: Initial step finds 'a' where \(a^2 \le 57\). We found \(a=7\). So the root is around 70-80. The remaining part is \(5776 - 70^2 = 5776 - 4900 = 876\). Now we look for a number 70+b such that \((70+b)^2 \approx 5776\). \((70+b)^2 = 70^2 + 2 \times 70 \times b + b^2 = 4900 + 140b + b^2\). We have \(4900 + 140b + b^2 = 5776\). So, \(140b + b^2 = 5776 - 4900 = 876\). This can be written as \(b(140+b) = 876\). In the long division step, doubling the root found so far (7) gives 14, and appending the next digit 'b' gives 14b. Multiplying by 'b' gives \(b \times (140+b)\) because the 14 represents \(2 \times 70\). We find \(b=6\), as \(6 \times (140+6) = 6 \times 146 = 876\). This confirms the long division method is an application of this algebraic principle.