Question

How do you find the value of $\sqrt{56}$ ?

Answer 1

Hint: We try to form the indices formula for the value 2. This is a square root of 56. We find the prime factorisation of 56. Then we take one digit out of the two same number of primes. There will be an odd number of primes remaining in the root which can’t be taken out. We keep the cube root in its simplest form.

Complete step-by-step solution:
We need to find the value of the algebraic form of $\sqrt{56}$. This is a square root form.
The given value is the form of indices. We are trying to find the root value of 56.
We know the theorem of indices \[{{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}\]. Putting value 2 we get \[{{a}^{\dfrac{1}{2}}}=\sqrt[2]{a}\].
We need to find the prime factorisation of the given number 56.
$\begin{align}
  & 2\left| \!{\underline {\,
  56 \,}} \right. \\
 & 2\left| \!{\underline {\,
  28 \,}} \right. \\
 & 2\left| \!{\underline {\,
  14 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Therefore, \[56=2\times 2\times 2\times 7\].
For finding the square root, we need to take one digit out of the two same number of primes.
This means in the cube root value of \[56=2\times 2\times 2\times 7\], we will take out one 2 from the multiplication.
So, $\sqrt{56}=\sqrt{2\times 2\times 2\times 7}=2\sqrt{14}$.
Therefore, the value of $\sqrt{56}$ is $2\sqrt{14}$.

Note: We can also use the variable form where we can take $x=\sqrt{56}$. But we need to remember that we can’t use the cube on both sides of the equation $x=\sqrt{56}$ as in that case we are taking two extra values as a root value. Then this linear equation becomes a cubic equation.