Question

What is the square root of 3 plus square root of 12?

Answer 1

Hint: To find the value of the square root of 3 plus square root of 12, we will first write the mathematical form of the statement. Then, we have to simplify $\sqrt{12}$ by prime factoring or using ladder method. Then, we have to substitute this value in the mathematical expression and add.

Complete step-by-step solution:
We have to find the value of the square root of 3 plus square root of 12. Let us express this statement mathematically.
$\Rightarrow \sqrt{3}+\sqrt{12}...\left( i \right)$
Let us simplify $\sqrt{12}$ using the ladder method.
$\begin{align}
  & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\text{ }1 \\
\end{align}$
Therefore, we can write $\sqrt{12}$ as $2\sqrt{3}$ . Now, let us substitute this value in (i).
$\Rightarrow \sqrt{3}+2\sqrt{3}$
Now, let us add the roots.
$\Rightarrow 3\sqrt{3}$
We can also convert the above value into its decimal form by substituting the value of $\sqrt{3}$ . We know that $\sqrt{3}=1.732$ .
$\Rightarrow 3\times 1.732=1.516$
Therefore, the square root of 3 plus the square root of 12 is $3\sqrt{3}$ or 1.516.

Note: Students must know to find the roots of values using either prime factorization method or ladder method. It is not necessary to convert the final answer into decimals. Students must know the square root of small numbers like 3, 2, 5, 6, 7, 8, 9 and 10. Students must note that we were able to add $\sqrt{3}$ and $2\sqrt{3}$ because the radicals (roots) are the same. If we had to add $\sqrt{3}$ and $3\sqrt{2}$ , we cannot directly add the roots. Instead, we will have to substitute the values of the roots and simplify.