Question

Please solve 17(iii)
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Answer 1

$\sf\small\underline\purple{Prove\:that\:-(i)}$

$\tt{\implies \bigg(\dfrac{x^a}{x^b}\bigg)^{(a+b)}\times\:\bigg(\dfrac{x^b}{x^c}\bigg)^{(b+c)}\times\:\bigg(\dfrac{x^c}{x^a}\bigg)^{(c+a)}=1}$

$\tt{\implies \dfrac{a^m}{a^n}=a^{m-n}}$

$\tt{\implies (x^{a-b})^{(a+b)}\times\:(x^{b-c})^{(b+c)}\times\:(x^{c-a})^{(c+a)}}$

$\tt{\implies (a^n)^m=(a)^{mn}}$

$\tt{\implies (x)^{(a-b)(a+b)}\times\:(x)^{(b-c)(b+c)}\times\:(x)^{(c-a)(c+a)}}$

$\tt{\implies a^2-b^2=(a+b)(a-b)}$

$\tt{\implies (x)^{a^{2}-b^{2}}\times\:(x)^{b^{2}-c^{2}}\times\:(x)^{c^{2}-a^{2}}}$

$\tt{\implies (x)^{(a^{2}-b^{2}+b^{2}-c^{2}+c^{2}-a^{2})}}$

$\tt{\implies (x)^{0}=1}$

$\sf\small\underline\purple{Prove\:that\:-(ii)}$

$\tt{\implies \bigg(\dfrac{x^a}{x^b}\bigg)^{(a+b-c)}\times\:\bigg(\dfrac{x^b}{x^c}\bigg)^{(b+c-a)}\times\:\bigg(\dfrac{x^c}{x^a}\bigg)^{(c+a-b)}=1}$

$\tt{\implies (x^{a-b})^{(a+b-c)}\times\:(x^{b-c})^{(b+c-a)}\times\:(x^{c-a})^{(c+a-b)}}$

$\tt{\implies (x)^{(a-b)(a+b-c)}\times\:(x)^{(b-c)(b+c-a)}\times\:(x)^{(c-a)(c+a-b)}}$

$\tt{\implies (x)^{(a^2+ab+ac-ab-b^2-ac)}.(x)^{(b^2+bc-ac-bc+ac-c^2)}.(x)^{(c^2+ac-bc-ac-a^2+bc)}}$

$\tt{\implies (x)^{a^{2}-b^{2}}\times\:(x)^{b^{2}-c^{2}}\times\:(x)^{c^{2}-a^{2}}}$

$\tt{\implies (x)^{(a^{2}-b^{2}+b^{2}-c^{2}+c^{2}-a^{2})}}$

$\tt{\implies (x)^{0}=1}$

Answer 2

1)

$\sf{:\implies (x^{a-b})^{(a+b)}\times\:(x^{b-c})^{(b+c)}\times\:(x^{c-a})^{(c+a)}}$

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

$\sf{:\implies (x)^{(a-b)(a+b)}\times\:(x)^{(b-c)(b+c)}\times\:(x)^{(c-a)(c+a)}}$

$\sf{:\implies (x)^{a^{2}-b^{2}}\times\:(x)^{b^{2}-c^{2}}\times\:(x)^{c^{2}-a^{2}}}$

$\sf{:\implies (x)^{(a^{2}-b^{2}+b^{2}-c^{2}+c^{2}-a^{2})}}$

$\bf{:\implies x^{0}=1}$

━━━━━━━━━━━━━━━━━━

2)

$\sf{:\implies (x^{a-b})^{(a+b-c)}\times\:(x^{b-c})^{(b+c-a)}\times\:(x^{c-a})^{(c+a-b)}}$

$\sf{:\implies (x)^{(a-b)(a+b-c)}\times\:(x)^{(b-c)(b+c-a)}\times\:(x)^{(c-a)(c+a-b)}}$

$\sf{:\implies (x)^{(a^2+ab+ac-ab-b^2-ac)}.(x)^{(b^2+bc-ac-bc+ac-c^2)} \times (x)^{(c^2+ac-bc-ac-a^2+bc)}}$

$\sf{:\implies (x)^{a^{2}-b^{2}}\times\:(x)^{b^{2}-c^{2}}\times\:(x)^{c^{2}-a^{2}}}$

$\sf{:\implies (x)^{(a^{2}-b^{2}+b^{2}-c^{2}+c^{2}-a^{2})}}$

$\bf{\implies x^{0}=1}$

━━━━━━━━━━━━━━━━━━