Prove that:

\( \left(\frac{x^{a}}{x^{-b}}\right)^{a^{2}-a b+b^{2}} \times\left(\frac{x^{b}}{x^{-c}}\right)^{b^{2}-b c+c^{2}} \times\left(\frac{x^{c}}{x^{-a}}\right)^{c^{2}-c a+a^{2}}=1 \)

Given:

\( \left(\frac{x^{a}}{x^{-b}}\right)^{a^{2}-a b+b^{2}} \times\left(\frac{x^{b}}{x^{-c}}\right)^{b^{2}-b c+c^{2}} \times\left(\frac{x^{c}}{x^{-a}}\right)^{c^{2}-c a+a^{2}}=1 \)

To do:

We have to prove that \( \left(\frac{x^{a}}{x^{-b}}\right)^{a^{2}-a b+b^{2}} \times\left(\frac{x^{b}}{x^{-c}}\right)^{b^{2}-b c+c^{2}} \times\left(\frac{x^{c}}{x^{-a}}\right)^{c^{2}-c a+a^{2}}=1 \).

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

$a^{m} \times a^{n}=a^{m+n}$

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$  

LHS $=(\frac{x^{a}}{x^{-b}})^{a^{2}-a b+b^{2}} \times(\frac{x^{b}}{x^{-c}})^{b^{2}-b c+c^{2}} \times(\frac{x^{c}}{x^{-a}})^{c^{2}-c a+a^{2}}$

$=(x^{a+b})^{a^{2}-a b+b^{2}} \times(x^{b+c})^{b^{2}-b c+c^{2}} \times(x^{c+a})^{c^{2}-c a+a^{2}}$

$=x^{(a+b)(a^{2}-a b+b^{2})} \times x^{(b+c)(b^{2}-b c+c^{2})} \times x^{(c+a)(c^{2}-c a+a^{2})}$

$=x^{a^{3}+b^{3}} \times x^{b^{3}+c^{3}} \times x^{c^{3}+a^{3}}$

$=x^{a^{3}+b^{3}+b^{3}+c^{3}+c^{3}+a^{3}}$

$=x^{2(a^{3}+b^{3}+c^{3})}$

$=$ RHS

Hence proved. 

Tutorialspoint

Updated on: 10-Oct-2022

39 Views

Kickstart Your Career

Get certified by completing the course

Get Started