Question

If $x=3-2\sqrt{2}$, then find ${{x}^{4}}-\dfrac{1}{{{x}^{4}}}$.

Answer 1

Hint: We will find the value of $\dfrac{1}{x}$ using the given value of $x$. We will find the values of ${{x}^{2}}$ and $\dfrac{1}{{{x}^{2}}}$. Then we will use the following law of indices, ${{a}^{m\times n}}={{\left( {{a}^{m}} \right)}^{n}}$. Using this law, we will find the value of ${{x}^{4}}$ and $\dfrac{1}{{{x}^{4}}}$. After finding these values, we will substitute them in the expression ${{x}^{4}}-\dfrac{1}{{{x}^{4}}}$ to obtain the desired answer.

Complete step by step answer:
We are given that $x=3-2\sqrt{2}$. We will find the value of $\dfrac{1}{x}$ in the following manner,
$\dfrac{1}{x}=\dfrac{1}{3-2\sqrt{2}}$
Now, we will multiply and divide the above expression by the conjugate of the denominator, that is, rationalize it. The conjugate of the denominator is $3+2\sqrt{2}$. Multiplying and dividing the above expression by $3+2\sqrt{2}$, we get
$\dfrac{1}{x}=\dfrac{1}{3-2\sqrt{2}}\times \dfrac{3+2\sqrt{2}}{3+2\sqrt{2}}$
In the denominator, we will use the algebraic identity $\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}$. So, we will obtain the following,
$\begin{align}
  & \dfrac{1}{x}=\dfrac{3+2\sqrt{2}}{{{\left( 3 \right)}^{2}}-{{\left( 2\sqrt{2} \right)}^{2}}} \\
 & \Rightarrow \dfrac{1}{x}=\dfrac{3+2\sqrt{2}}{9-8} \\
 & \therefore \dfrac{1}{x}=3+2\sqrt{2} \\
\end{align}$
Now we have $x=3-2\sqrt{2}$ and $\dfrac{1}{x}=3+2\sqrt{2}$.
Next, we will find the values of ${{x}^{2}}$ and $\dfrac{1}{{{x}^{2}}}$ as follows,
$\begin{align}
  & {{x}^{2}}={{\left( 3-2\sqrt{2} \right)}^{2}} \\
 & \Rightarrow {{x}^{2}}={{3}^{2}}+{{\left( 2\sqrt{2} \right)}^{2}}-\left( 2\times 3\times 2\sqrt{2} \right) \\
 & \Rightarrow {{x}^{2}}=9+8-12\sqrt{2} \\
 & \therefore {{x}^{2}}=17-12\sqrt{2} \\
\end{align}$
Now for $\dfrac{1}{{{x}^{2}}}$ we get the following,
$\begin{align}
  & \dfrac{1}{{{x}^{2}}}={{\left( \dfrac{1}{x} \right)}^{2}} \\
 & \Rightarrow \dfrac{1}{{{x}^{2}}}={{\left( 3+2\sqrt{2} \right)}^{2}} \\
 & \Rightarrow \dfrac{1}{{{x}^{2}}}={{3}^{2}}+{{\left( 2\sqrt{2} \right)}^{2}}+\left( 2\times 3\times 2\sqrt{2} \right) \\
 & \Rightarrow \dfrac{1}{{{x}^{2}}}=9+8+12\sqrt{2} \\
 & \therefore \dfrac{1}{{{x}^{2}}}=17+12\sqrt{2} \\
\end{align}$
We have to find the value of ${{x}^{4}}-\dfrac{1}{{{x}^{4}}}$. So, we need to find the values of ${{x}^{4}}$ and $\dfrac{1}{{{x}^{4}}}$. We know that ${{a}^{m\times n}}={{\left( {{a}^{m}} \right)}^{n}}$. This implies that \[{{x}^{4}}={{x}^{2\times 2}}={{\left( {{x}^{2}} \right)}^{2}}\] and $\dfrac{1}{{{x}^{4}}}={{\left( \dfrac{1}{x} \right)}^{4}}={{\left( {{\left( \dfrac{1}{x} \right)}^{2}} \right)}^{2}}={{\left( \dfrac{1}{{{x}^{2}}} \right)}^{2}}$. Using this, we will calculate the values of ${{x}^{4}}$ and $\dfrac{1}{{{x}^{4}}}$ in the following manner,
$\begin{align}
  & {{x}^{4}}={{\left( {{x}^{2}} \right)}^{2}} \\
 & \Rightarrow {{x}^{4}}={{\left( 17-12\sqrt{2} \right)}^{2}} \\
 & \Rightarrow {{x}^{4}}={{17}^{2}}+{{\left( 12\sqrt{2} \right)}^{2}}-\left( 2\times 17\times 12\sqrt{2} \right) \\
 & \Rightarrow {{x}^{4}}=289+288-408\sqrt{2} \\
 & \therefore {{x}^{4}}=577-408\sqrt{2} \\
\end{align}$
Similarly, for $\dfrac{1}{{{x}^{4}}}$ we get
$\begin{align}
  & \dfrac{1}{{{x}^{4}}}={{\left( \dfrac{1}{{{x}^{2}}} \right)}^{2}} \\
 & \Rightarrow \dfrac{1}{{{x}^{4}}}={{\left( 17+12\sqrt{2} \right)}^{2}} \\
 & \Rightarrow \dfrac{1}{{{x}^{4}}}={{17}^{2}}+{{\left( 12\sqrt{2} \right)}^{2}}+\left( 2\times 17\times 12\sqrt{2} \right) \\
 & \Rightarrow \dfrac{1}{{{x}^{4}}}=289+288+408\sqrt{2} \\
 & \therefore \dfrac{1}{{{x}^{4}}}=577+408\sqrt{2} \\
\end{align}$
Now, we will find the value of ${{x}^{4}}-\dfrac{1}{{{x}^{4}}}$ in the following manner,
$\begin{align}
  & {{x}^{4}}-\dfrac{1}{{{x}^{4}}}=577-408\sqrt{2}-\left( 577+408\sqrt{2} \right) \\
 & \Rightarrow {{x}^{4}}-\dfrac{1}{{{x}^{4}}}=577-408\sqrt{2}-577-408\sqrt{2} \\
 & \therefore {{x}^{4}}-\dfrac{1}{{{x}^{4}}}=-816\sqrt{2} \\
\end{align}$

So, the required answer is ${{x}^{4}}-\dfrac{1}{{{x}^{4}}}=-816\sqrt{2}$.

Note: It is essential that we are familiar with the laws of indices in this type of questions. The calculations in this question are tricky and there is a possibility to make errors in such calculations. So, it is useful if we write the calculations in an explicit manner. We should also be careful with the signs in all the expressions.