Question

Find the square roots of the given number by Prime factorization method: $7056$.

Answer 1

Hint: Prime factorization method is a method in which we use the product of only prime numbers to form the value given in the question and then we do a square root of that product of prime numbers and then we find the square-root of the number in terms of the product of prime numbers. To find the prime numbers we use the formula of LCM on the number.

Complete step-by-step answer:
We find the LCM of the given number in its lowest form or prime numbers.
\[\begin{align}
  & 2\left| 7056 \right. \\
 & 2\left| 3528 \right. \\
 & 2\left| 1764 \right. \\
 & 2\left| 882 \right. \\
 & 3\left| 441 \right. \\
 & 3\left| 147 \right. \\
 & 7\left| 49 \right. \\
 & 7\left| 7 \right. \\
 & 1 \\
\end{align}\]
Now the product of prime numbers are collapsed in form of power product of prime numbers giving us the final product as \[2\times 2\times 2\times 2\times 3\times 3\times 7\times 7={{2}^{4}}\times {{3}^{2}}\times {{7}^{2}}\].
Now square root the prime number product we get:
Square root of \[7056={{2}^{4}}\times {{3}^{2}}\times {{7}^{2}}\].
\[=\sqrt{{{2}^{4}}\times {{3}^{2}}\times {{7}^{2}}}\]
\[={{2}^{2}}\times 3\times 7\]
\[=84\]
Hence, the square root of \[7056\] is \[84\].

Note: Students may go wrong if they try to find the square root by using shortcut method by checking the unit digit of the number asked in the question and then find the value accordingly although LCM process is length but the question has asked prime factorization hence, first finding the answer by shortcut and then matching it with the method will only take time hence, first LCM and then product of the prime numbers by reducing half their power is the correct way to go.