- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- MS Excel
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies
- Legal Studies
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
$x+\frac{1}{x}=4$ find the value of the following:
a) $x^{2}+\frac{1}{x^{2}}$
b) $x^{4}+\frac{1}{x 4}$
Given :
The given expression is $x+\frac{1}{x}=4$.
To do :
We have to find the values of,
a) $x^{2}+\frac{1}{x^{2}}$ b) $x^{4}+\frac{1}{x 4}$.
Solution :
a) $x^{2}+\frac{1}{x^{2}}$
$x+\frac{1}{x}=4$
Squaring on both sides,
$(x+\frac{1}{x})^2=4^2$
$x^2 + \frac{1}{x^2} +2.x.\frac{1}{x} = 16$ $[(a+b)^2=a^2+b^2+2ab]$
$x^2 + \frac{1}{x^2} +2=16$
$x^2 + \frac{1}{x^2} =16-2$
$x^2 + \frac{1}{x^2} =14$.
Therefore, the value of $x^2 + \frac{1}{x^2}$ is 14.
b) $x^{4}+\frac{1}{x 4}$
We know that, $x^2 + \frac{1}{x^2} =14$
Squaring on both sides,
$(x^2 + \frac{1}{x^2})^2 =14^2$
$(x^2)^2 + (\frac{1}{x^2})^2 +2 .x^2.\frac{1}{x^2}=14^2$ $[(a+b)^2=a^2+b^2+2ab]$
$x^4 + \frac{1}{x^4} + 2 = 196$
$x^4 + \frac{1}{x^4}=196-2$
$x^4 + \frac{1}{x^4}=194$.
Therefore, the value of $x^4 + \frac{1}{x^4}$ is 194.