Write the following in the expanded form:$ (\frac{x}{y}+\frac{y}{z}+\frac{z}{x})^{2} $
Given:
\( (\frac{x}{y}+\frac{y}{z}+\frac{z}{x})^{2} \)
To do:
We have to write the given expression in expanded form.
Solution:
We know that,
$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$
Therefore,
$(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})^{2}=(\frac{x}{y})^{2}+(\frac{y}{z})^{2}+(\frac{z}{x})^{2}+2 \times \frac{x}{y} \times \frac{y}{z}+2 \times \frac{y}{z} \times \frac{z}{x}+2 \times \frac{z}{x} \times \frac{x}{y}$
$=\frac{x^{2}}{y^{2}}+\frac{y^{2}}{z^{2}}+\frac{z^{2}}{x^{2}}+2 \frac{x}{z}+2 \frac{y}{x}+2 \frac{z}{y}$
Hence, $(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})^{2}=\frac{x^{2}}{y^{2}}+\frac{y^{2}}{z^{2}}+\frac{z^{2}}{x^{2}}+2 \frac{x}{z}+2 \frac{y}{x}+2 \frac{z}{y}$.
Related Articles
- Find the following products:\( \frac{-4}{27} x y z\left[\frac{9}{2} x^{2} y z-\frac{3}{4} x y z^{2}\right] \)
- Simplify each of the following expressions:\( (x+y+z)^{2}+\left(x+\frac{y}{2}+\frac{z}{3}\right)^{2}-\left(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}\right)^{2} \)
- Find the following products:\( \frac{-8}{27} x y z\left(\frac{3}{2} x y z^{2}-\frac{9}{4} x y^{2} z^{3}\right) \)
- Write the following in the expanded form:\( (2 x-y+z)^{2} \)
- Write the following in the expanded form:\( (-3 x+y+z)^{2} \)
- Verify associativity of addition of rational numbers i.e., $(x + y) + z = x + (y + z)$, when:(i) \( x=\frac{1}{2}, y=\frac{2}{3}, z=-\frac{1}{5} \)(ii) \( x=\frac{-2}{5}, y=\frac{4}{3}, z=\frac{-7}{10} \)(iii) \( x=\frac{-7}{11}, y=\frac{2}{-5}, z=\frac{-3}{22} \)(iv) \( x=-2, y=\frac{3}{5}, z=\frac{-4}{3} \)
- Verify the property \( x \times(y+z)=(x \times y)+(x \times z) \) for the given values of \( x,\ y \) and \( z \).\( x=\frac{-5}{2}, y=\frac{1}{2} \) and \( z=-\frac{10}{7} \)>
- If \( 2^{x}=3^{y}=12^{z} \), show that \( \frac{1}{z}=\frac{1}{y}+\frac{2}{x} \).
- Write the following in the expanded form:\( (x+2 y+4 z)^{2} \)
- Verify the property: $x \times(y + z) = x \times y + x \times z$ by taking:(i) \( x=\frac{-3}{7}, y=\frac{12}{13}, z=\frac{-5}{6} \)(ii) \( x=\frac{-12}{5}, y=\frac{-15}{4}, z=\frac{8}{3} \)(iii) \( x=\frac{-8}{3}, y=\frac{5}{6}, z=\frac{-13}{12} \)(iv) \( x=\frac{-3}{4}, y=\frac{-5}{2}, z=\frac{7}{6} \)
- Write the following in the expanded form:\( (-2 x+3 y+2 z)^{2} \)
- Verify that \( x^{3}+y^{3}+z^{3}-3 x y z=\frac{1}{2}(x+y+z)\left[(x-y)^{2}+(y-z)^{2}+(z-x)^{2}\right] \)
- Find the following product.$\frac{1}{2} x y \times \frac{2}{3} x^{2} y z^{2}$
- Verify: $x\times(y\times z)=(x\times y)\times z$, where $x=\frac{1}{2},\ y=\frac{1}{3}$ and $z=\frac{1}{4}$.
- Verify the property: $x \times (y \times z) = (x \times y) \times z$ by taking:(i) \( x=\frac{-7}{3}, y=\frac{12}{5}, z=\frac{4}{9} \)(ii) \( x=0, y=\frac{-3}{5}, z=\frac{-9}{4} \)(iii) \( x=\frac{1}{2}, y=\frac{5}{-4}, z=\frac{-7}{5} \)(iv) \( x=\frac{5}{7}, y=\frac{-12}{13}, z=\frac{-7}{18} \)
Kickstart Your Career
Get certified by completing the course
Get Started
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