Question

How do you solve $\sqrt {\left( {2x} \right)} = 6$ ?

Answer 1

Hint: In order to solve and write the expression into the simplest form . The square root is related to figuring out what should be the number which when multiplied by itself is equal to the number under the square root symbol $\sqrt {} $. This symbol is known as radical. Since in our case we have given the question in which we have to solve and find the value of x , we will first get rid of the radical and remove the square root as we want the original value of x , by somewhere using equivalent equations. Equivalent equations are said to be algebraic equations that may have the same solutions if we add or subtract the same number to both sides of an equation - Left hand side or Right hand side of the equal to sign.

Complete step-by-step solution:
If we see the question , we need to solve the given expression under the square root which is $\sqrt {\left( {2x} \right)} = 6$ .
By applying the concept of equivalent equation , we will first subtract the number 6 from both the L . H . S . and the R . H . S . as follows –
$\sqrt {\left( {2x} \right)} - 6 = 0$
Add the number 6 to both the L . H . S . and the R . H . S . as follows –
$\sqrt {\left( {2x} \right)} = 6$
For getting rid of the radical on the L . H . S . of the equation , we need to square on both sides of the equation .
${\left( {\sqrt {\left( {2x} \right)} } \right)^2} = {6^2}$
This radical can also be rewritten as raised to the power of $\dfrac{1}{2}$ which makes the square and the radical unity the power will become 1 which will pull out of the radical . We will pull out the number 2 raised to the power two when pulled out will make the power unity that is 1.
The formula used is the power rule and which multiply the exponents as follows –
\[{\left( {{a^m}} \right)^n} = {a^{mn}}\]
Now , as per the mathematical statement of the question we have to solve the square root .
Applying the formula gives ,
So , we will cancel out the common factor 2 in the power and rewrite the expression ,
\[{(2x)^1} = {6^2}\]
Now , we are going to simplify the expression further ,
As 6 raised to the power of two is regarded as square or the number 36 which itself is a perfect square by the definition of square root that states the number which when multiplied by itself is equal to the number under the square root is being implemented and satisfied .
Which results in
\[2x = 36\]
Divide both the L . H . S . and the R . H . S . by 2 ,
\[
  x = \dfrac{{36}}{2} \\
  x = 18 \\
 \]
Therefore , the final answer is \[x = 18\].

Note: We can use prime factorisation for the number inside the radical and pull out non- radical terms or perfect squares from the inside of the square root to make the solution easier. Try to factorize the base part of the value inside the square root such that it contains perfect squares in it. We will pull out the number 2 raised to the power two when pulled out will make the power unity that is 1 .
Cross check the answer and always keep the final answer simplified.