Question

What is the value of $\sqrt{126}+\sqrt{56}$ in standard form?

Answer 1

Hint: Consider the two square root terms one by one and write them as the product of their prime factors. Now, try to form groups to similar prime factors and leave the factors which cannot be grouped. For the grouped factors use the formula of exponents ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to simplify the radical expression. Now, add the two terms if in the end you get the same numerical value inside the radical sign.

Complete step by step solution:
Here we have been provided with the expression $\sqrt{126}+\sqrt{56}$ and we are asked to write its standard form. That means we need to simplify it and write the simplest radical form. Let us assume this expression as E, so we have,
$\Rightarrow E=\sqrt{126}+\sqrt{56}$
Now, to simplify the above expression further we need to write the numbers present inside the radical as the product of their prime factors. Since, we have the under root sign in the radical expression so we will try to form pairs of two similar prime factors if they will appear and we will use the formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to remove them from the radical sign. The factors which will appear only once will be left inside the radical sign. So we get,
$\Rightarrow 126=2\times 3\times 3\times 7={{3}^{2}}\times 2\times 7$
$\Rightarrow 56=2\times 2\times 2\times 7={{2}^{2}}\times 2\times 7$
Substituting the above values in the expression E we get,
$\Rightarrow E=\sqrt{{{3}^{2}}\times 2\times 7}+\sqrt{{{2}^{2}}\times 2\times 7}$
Using the formula of exponents ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ for the paired factors we get,
\[\begin{align}
  & \Rightarrow E=\left( {{3}^{2\times \dfrac{1}{2}}}\times \sqrt{2\times 7} \right)+\left( {{2}^{2\times \dfrac{1}{2}}}\times \sqrt{2\times 7} \right) \\
 & \Rightarrow E=3\sqrt{14}+2\sqrt{14} \\
 & \therefore E=5\sqrt{14} \\
\end{align}\]
Hence, the simplified form of the given expression is $5\sqrt{14}$.

Note: You may note that here you do not have to find the square root but only simplify it so do not try to use the long division method to get the answer in decimal form. Remember all the formulas of exponents and powers as they are frequently used in other chapters of mathematics. Note that in the end we got $\sqrt{14}$ as the radical expression in both the terms and that is why we were able to add 3 and 2 present outside the radical sign.