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Definition of Functions

xax-ba2-ab+b2...

Question

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

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Solution

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

On further simplification, we have:

$x^{(a + b) (a^{2} - a b + b^{2})} \times x^{(b + c) (b^{2} - b c + c^{2})} \times x^{(c + a) (c^{2} - c a + a^{2})}$
$[∵ \frac{1}{a^{- x}} = a^{x}]$

$= x^{a^{3} + b^{3}} . x^{b^{3} + c^{3}} . x^{c^{3} + a^{3}}$
$[∵ (x + y) (x^{2} - x y + y^{2}) = x^{3} + y^{3}]$

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

Hence, simplified value is $x^{2 (a^{3} + b^{3} + c^{3})}$

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Similar questions

Q. Prove that:

(i)

{(\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}} = 1

{(\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}} = 1

{(\frac{x^{a}}{x^{b}})}^{c} \times {(\frac{x^{b}}{x^{c}})}^{a} \times {(\frac{x^{c}}{x^{a}})}^{b} = 1

Prove that :

(i) ${(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \times {(\frac{x^{b}}{x^{c}})}^{b^{2} + a b + c^{2}} \times {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + a^{2}} = 1$

(ii) ${(\frac{x^{a}}{x^{b}})}^{c} \times {(\frac{x^{b}}{x^{c}})}^{a} \times {(\frac{x^{c}}{x^{a}})}^{b} = 1$

Write the following by using ∑ notation

(1) ab(a + b) + bc(b + c) + ca(c + a)

(2) a² + b² + c² − ab − bc − ca

(3) x(y² + z²) + y(z² − x²) + z(x² − y²)

(4) ab² − ac² − a²b + bc² − cb² + ca²

Q. Prove that :

|\begin{matrix} 1 & b + c & b^{2} + c^{2} \\ 1 & c + a & c^{2} + a^{2} \\ 1 & a + b & a^{2} + b^{2} \end{matrix}| = (a - b) (b - c) (c - a)

State true or false:

(\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^{2 (a^{3} + b^{3} + c^{3})}

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Definition of Functions

Standard XII Mathematics

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