Question

How do you solve $ 3{{x}^{\dfrac{1}{4}}}=4 $ ?

Answer 1

Hint: We have to find the value of x in this question, so we solve this problem by using Exponential identities. Thus, we start solving this problem by dividing both sides by 3 and after necessary calculation, we eliminate the fractional exponent in LHS by raising the power 4 on both sides of the equation. Then, we will use exponential identity $ {{({{x}^{\dfrac{1}{a}}})}^{b}}={{x}^{\dfrac{1}{a}.b}}={{x}^{\dfrac{b}{a}}} $ on the left-hand side of the equation and will make the necessary calculations to get the value of x, which is our required answer.

Complete step by step answer:
According to the question, we have to find the value of x.
The equation is $ 3{{x}^{\dfrac{1}{4}}}=4 $ ---------- (1)
First, we divide both sides of the equation (1) by 3, we get
 $ \Rightarrow \dfrac{3}{3}{{x}^{\dfrac{1}{4}}}=\dfrac{4}{3} $
As we know, the same terms will cancel out in the division, thus we get
 $ \Rightarrow {{x}^{\dfrac{1}{4}}}=\dfrac{4}{3} $
In the above equation, we see that LHS has a fractional power $ \dfrac{1}{4} $ , so we will try eliminating the fractional exponent to the natural exponent.
Therefore, we will raise both sides of the equation to the power 4, we get
 $ \Rightarrow {{\left( {{x}^{\dfrac{1}{4}}} \right)}^{4}}={{\left( \dfrac{4}{3} \right)}^{4}} $
Now, we will use exponential identity $ {{({{x}^{\dfrac{1}{a}}})}^{b}}={{x}^{\dfrac{1}{a}.b}}={{x}^{\dfrac{b}{a}}} $ in the LHS of the above equation, we get
 $ \Rightarrow ~{{x}^{\dfrac{1}{4}.4}}={{\left( \dfrac{4}{3} \right)}^{4}} $
Thus, on further solving the above equation, we get
 $ \begin{align}
  & \Rightarrow {{x}^{1}}={{\left( \dfrac{4}{3} \right)}^{4}} \\
 & \Rightarrow x=\dfrac{256}{81} \\
\end{align} $
Therefore, for the equation $ 3{{x}^{\dfrac{1}{4}}}=4 $ , the value of x is $ \dfrac{256}{81} $ .

Note:
 Always keep in mind, how to eliminate the fractional exponent in this question. Always multiply the reciprocal of the number, to get a natural number 1 in the exponent of the variable. One of the alternative methods to solve this problem is to raise both sides to power 4, and then apply exponential identities $ {{(xy)}^{a}}={{x}^{a}}.{{y}^{a}} $ and $ {{({{x}^{\dfrac{1}{a}}})}^{b}}={{x}^{\dfrac{1}{a}.b}}={{x}^{\dfrac{b}{a}}} $ , make necessary calculations, to get the value of x, which is our required answer.
An alternative method:
Equation: $ 3{{x}^{\dfrac{1}{4}}}=4 $
Firstly, we will raise both sides of the equation to the power 4 in equation (2), we get
 $ \Rightarrow {{\left( 3{{x}^{\dfrac{1}{4}}} \right)}^{4}}={{4}^{4}} $
Now, we will apply exponential identities $ {{(xy)}^{a}}={{x}^{a}}.{{y}^{a}} $ and $ {{({{x}^{\dfrac{1}{a}}})}^{b}}={{x}^{\dfrac{1}{a}.b}}={{x}^{\dfrac{b}{a}}} $ in the LHS of the above equation, we get
 $ \Rightarrow {{3}^{4}}.\left( {{x}^{\dfrac{1}{4}.4}} \right)={{4}^{4}} $
Therefore, on further calculations, we get
 $ \Rightarrow 81.x=256 $
Now, we will divide both sides by 81, we get
  $ \Rightarrow \dfrac{81}{81}.x=\dfrac{256}{81} $
Thus, we get
 $ \Rightarrow x=\dfrac{256}{81} $ which is our required answer.