Question

(x^a/x^b)^c × (x^b/x^c)^a × (x^c/x^a) ^a

​

Answer 1

CORRECT QUESTION:

Simplify:

$\left( \dfrac{x^a}{x^b} \right)^c \times \left(\dfrac{x^b}{x^c}\right)^a \times \left(\dfrac{x^c}{x^a}\right) ^b$

ANSWER:

$\left( \dfrac{x^a}{x^b} \right)^c \times \left(\dfrac{x^b}{x^c}\right)^a \times \left(\dfrac{x^c}{x^a}\right) ^b = 1$

GIVEN:

$\left( \dfrac{x^a}{x^b} \right)^c \times \left(\dfrac{x^b}{x^c}\right)^a \times \left(\dfrac{x^c}{x^a}\right) ^b$

TO SIMPLIFY:

$\left( \dfrac{x^a}{x^b} \right)^c \times \left(\dfrac{x^b}{x^c}\right)^a \times \left(\dfrac{x^c}{x^a}\right) ^b$

EXPLANATION:

$\left( \dfrac{x^a}{x^b} \right)^c \times \left(\dfrac{x^b}{x^c}\right)^a \times \left(\dfrac{x^c}{x^a}\right) ^b$

$\left( \dfrac{x^{ac}}{x^{bc}} \right) \times \left(\dfrac{x^{ab}}{x^{ac}}\right)\times \left(\dfrac{x^{bc}}{x^{ab}}\right)$

$\left( \dfrac{x^{ac} \times x^{ab} \times x^{bc}}{x^{bc} \times x^{ac} \times x^{ab}} \right)$

BOTH NUMERATOR AND DENOMINATOR ARE SAME. HENCE

$\left( \dfrac{x^{ac} \times x^{ab} \times x^{bc}}{x^{bc} \times x^{ac} \times x^{ab}} \right) = 1$

$\left( \dfrac{x^a}{x^b} \right)^c \times \left(\dfrac{x^b}{x^c}\right)^a \times \left(\dfrac{x^c}{x^a}\right) ^b = 1$

SOME FORMULAE:

$\left(X^a\right)^n= X^{an}$

$\left(\dfrac{X^m}{X^n}\right)^n= X^{m-n}$

$\left({X^m}{X^n}\right)= X^{m+n}$

$(X^0) = 1$

$(0^n) = 0$