Question

(xa+b)2(xb+c)2(xc+a)2÷(xaxbxc)4=1

Answer 1

$\underline{\textsf{To prove:}}$

$\mathsf{\dfrac{(x^{a+b})^2(x^{b+c})^2(x^{c+a})^2}{(x^a\,x^b\,x^c)^4}=1}$

$\underline{\textsf{Solution:}}$

$\textsf{Consider,}$

$\mathsf{\dfrac{(x^{a+b})^2(x^{b+c})^2(x^{a+b})^2}{(x^a\,x^b\,x^c)^4}}$

$\mathsf{=\dfrac{x^{2(a+b)}\,x^{2(b+c)}\,x^{2(a+b)}}{(x^{a+b+c})^4}}$

$\mathsf{=\dfrac{x^{2a+2b}\,x^{2b+2c}\,x^{2c+2a}}{(x^{a+b+c})^4}}$

$\mathsf{=\dfrac{x^{2a+2b+2b+2c+2c+2a}}{x^{4(a+b+c)}}}$

$\mathsf{=\dfrac{x^{4a+4b+4c}}{x^{4a+4b+4c}}}$

$\mathsf{=1}$

$\implies\boxed{\mathsf{\dfrac{(x^{a+b})^2(x^{b+c})^2(x^{c+a})^2}{(x^a\,x^b\,x^c)^4}=1}}$

Find more:

Simplify 73*73*73+27*27*27/73*73-73*27+27*27

https://brainly.in/question/2079320

Simplify [p^(a-b)]^c x [p^(a-c)]^b x [p^(-a)]^a x [p^b]^c

https://brainly.in/question/3922554

Answer 2

SOLUTION

TO PROVE

$\sf{} {( {x}^{a + b} )}^{2}. {( {x}^{b + c} )}^{2} . {( {x}^{b + c} )}^{2} \div {( {x}^{a} . {x}^{b} . {x}^{c} )}^{4} = 1$

FORMULA TO BE IMPLEMENTED

We are aware of the formula of indices that

$\sf{}1. \: \: {a}^{m} . {a}^{n} = {a}^{m + n}$

$\sf{}2. \: \: {( {a}^{m}) }^{n} = {a}^{mn}$

$\displaystyle \sf{}3. \: \: \: \frac{ { a}^{m} }{ {a}^{n} } = {a}^{m - n}$

EVALUATION

$\sf{} {( {x}^{a + b} )}^{2}. {( {x}^{b + c} )}^{2} . {( {x}^{b + c} )}^{2} \div {( {x}^{a} . {x}^{b} . {x}^{c} )}^{4}$

$= \displaystyle \sf{} \frac{ {( {x}^{a + b} )}^{2}. {( {x}^{b + c} )}^{2} . {( {x}^{b + c} )}^{2}}{{( {x}^{a} . {x}^{b} . {x}^{c} )}^{4} }$

$= \displaystyle \sf{} \frac{ {( {x}^{2a + 2b} )}. {( {x}^{2b + 2c} )}. {( {x}^{2b + 2c} )}}{{ {x}^{4a} . {x}^{4b} . {x}^{4c} } } \: ( \: by \: formula \: 2 \: )$

$= \displaystyle \sf{} \frac{ {x}^{2a + 2b + 2b + 2c + 2c + 2a} }{ {x}^{4a + 4b + 4c} } \: \: (by \: formula \: 2)$

$= \displaystyle \sf{} \frac{ {x}^{4a + 4b + 4c} }{ {x}^{4a + 4b + 4c} }$

$\displaystyle \sf{} = {x}^{(4a + 4b + 4c) - (4a + 4b + 4c)} \: \: ( \: by \: formula \: 3)$

$\displaystyle \sf{} = {x}^{0}$

$\sf{} = 1$

Hence proved

━━━━━━━━━━━━━━━━

LEARN MORE FROM BRAINLY

Let r be a positive number satisfying

r^1/1234) + (r^-1/1234) = 2

then find (r^4321)+(r^-4321)

https://brainly.in/question/22386838