Show that - xa-x-b a-b-xb-x-c b-c-xc-x-ac-1-1

L.H.S. (x^a/x^ b )^a b.(x^b/x^ c )^b c.(x^c/x^ a)^c 1 =(x^a+b)^a b.(x^b+c)^b c.(x^c+a)^c a =x^(a+b)(a b).x^(b+c)(b c).x^(c+a)(c a) =x^a^2 b^2+b^2 c^2+c^2 a^2 ...

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Byju's Answer

Standard VIII

Mathematics

Power of a Power

Show that : x...

Question

Show that :

${(\frac{x^{a}}{x^{- b}})}^{a - b} . {(\frac{x^{b}}{x^{- c}})}^{b - c} . {(\frac{x^{c}}{x^{- a}})}^{c - 1} = 1$

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Solution

$L.H.S. {(\frac{x^{a}}{x^{- b}})}^{a - b} . {(\frac{x^{b}}{x^{- c}})}^{b - c} . {(\frac{x^{c}}{x^{- a}})}^{c - 1} = (x^{a + b})^{a - b} . (x^{b + c})^{b - c} . (x^{c + a})^{c - a} = x^{(a + b) (a - b)} . x^{(b + c) (b - c)} . x^{(c + a) (c - a)} = x^{a^{2} - b^{2} + b^{2} - c^{2} + c^{2} - a^{2}} = x^{0} = 1 = R.H.S.$

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Similar questions

Simplify :

$(x^{a + b})^{a - b} . (x^{b + c})^{b - c} . (x^{c + 1})^{c - a}$

Q. The value of

\frac{x^{a + b} \cdot x^{b + c} \cdot x^{c + a}}{(x^{a} \cdot x^{b} \cdot x^{c})^{2}}

Q. Show that

\int \sqrt{(\frac{x + 1}{x - 1})} d x = \sqrt{x^{2} - 1} + log [x + \sqrt{x^{2} - 1}]

Q. Show that

\lim_{x \to \infty} (\sqrt{x^{2} + x + 1} - x) \neq \lim_{x \to \infty} (\sqrt{x^{2} + 1} - x)

Show that ${lim}_{x \to \infty} (\sqrt{x^{2} + x + 1 - x}) \neq {lim}_{x \to \infty} (\sqrt{x^{2} + 1} - x)$

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Show that : (xax−b)a−b.(xbx−c)b−c.(xcx−a)c−1=1

L.H.S.(xax−b)a−b.(xbx−c)b−c.(xcx−a)c−1=(xa+b)a−b.(xb+c)b−c.(xc+a)c−a=x(a+b)(a−b).x(b+c)(b−c).x(c+a)(c−a)=xa2−b2+b2−c2+c2−a2=x0=1=R.H.S.

Show that :

${(\frac{x^{a}}{x^{- b}})}^{a - b} . {(\frac{x^{b}}{x^{- c}})}^{b - c} . {(\frac{x^{c}}{x^{- a}})}^{c - 1} = 1$