If a=x^ m+n y^l, b=x^ n+l y^m and c=x^ l+my^n , prove that a^ m-n b^ n-l c^ l-m =1.

a=x^ m+n y^l, b=x^ n+l y^m and c=x^ l+my^n ,Now, LHS=a^ m-n b^ n-l c^ l-m = (x^ m+n y^l )^ m-n (x^ n+l )^ n-l (x^ l+m y^n )^ l-m =x^ (m+n)(m-n) ^ l(m-n) ^ (n+l)(n-l) ^ m(n-l) ^ (l+m)(l-m) ^ n(l-m) =x^ m^2-n^2 ^ lm-nl ^ n^2-l^2 ^ mn-lm ^ l^2-m^2 ^ nl-mn =x^ m^2-n^2+n^2-l^2+l^2-m^2 ^ lm-nl+mn-lm+nl-mn =x^0 ^0 =1 1 =1 =RHS

If a=x^ m+n y^l, b=x^ n+l y^m and c=x^ l+my^n , prove that a^ m-n…

Download App

Question

If $\text{a}=\text{x}^{\text{m}+\text{n}}\text{y}^\text{l},\ \text{b}=\text{x}^{\text{n}+\text{l}}\text{y}^\text{m}$ and $\text{c}=\text{x}^{\text{l}+\text{m}\text{y}^\text{n}},$ prove that $\text{a}^{\text{m}-\text{n}}\text{b}^{\text{n}-\text{l}}\text{c}^{\text{l}-\text{m}}=1.$

✓

Answer

$\text{a}=\text{x}^{\text{m}+\text{n}}\text{y}^\text{l},\ \text{b}=\text{x}^{\text{n}+\text{l}}\text{y}^\text{m}$ and $\text{c}=\text{x}^{\text{l}+\text{m}\text{y}^\text{n}},$Now,
$\text{LHS}=\text{a}^{\text{m}-\text{n}}\text{b}^{\text{n}-\text{l}}\text{c}^{\text{l}-\text{m}}$
$=\big(\text{x}^{\text{m}+\text{n}}\text{y}^\text{l}\big)^{\text{m}-\text{n}}\times\big(\text{x}^{\text{n}+\text{l}}\big)^{\text{n}-\text{l}}\times\big(\text{x}^{\text{l}+\text{m}}\text{y}^\text{n}\big)^{\text{l}-\text{m}}$
$=\text{x}^{(\text{m}+\text{n})(\text{m}-\text{n})}\times\text{y}^{\text{l}(\text{m}-\text{n})}\times\text{x}^{(\text{n}+\text{l})(\text{n}-\text{l})}\times\text{y}^{\text{m}(\text{n}-\text{l})}\times\text{x}^{(\text{l}+\text{m})(\text{l}-\text{m})}\times\text{y}^{\text{n}(\text{l}-\text{m})}$
$=\text{x}^{\text{m}^2-\text{n}^2}\times\text{y}^{\text{lm}-\text{nl}}\times\text{x}^{\text{n}^2-\text{l}^2}\times\text{y}^{\text{mn}-\text{lm}}\times\text{x}^{\text{l}^2-\text{m}^2}\times\text{y}^{\text{nl}-\text{mn}}$
$=\text{x}^{\text{m}^2-\text{n}^2+\text{n}^2-\text{l}^2+\text{l}^2-\text{m}^2}\times\text{y}^{\text{lm}-\text{nl}+\text{mn}-\text{lm}+\text{nl}-\text{mn}}$
$=\text{x}^0\times\text{y}^0$
$=1\times1$
$=1$
$=\text{RHS}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "2 Mark Question" from Exponents Of Real Numbers→Exponents Of Real Numbers — all question types→Maths STD 9 question papers→Maharashtra Board STD 9 question papers→Maharashtra Board question paper generator→

Exponents Of Real Numbers — Maths STD 9 — Question

Answer

Need a full question paper?

Explore more

Similar questions