Question

assuming that x is a positive real number and a b c are rational numbers show that (x^b/x^c)^a(x^c/x^a)^b(x^a/x^b)^c=1​

Answer 1

Step-by-step explanation:

Given :-

x is a positive real number and a, b, c are rational numbers .

To find :-

Show that (x^b/x^c)^a(x^c/x^a)^b(x^a/x^b)^c=1

Solution:-

Given that :

x is a positive real number .

a, b, c are rational numbers .

LHS:-

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

=> [x^(b-c)]^a × [x^(c-a)]^b × [x^(a-b]^c

Since a^m/a^n = a^(m-n)

=> x^a(b-c) × x^b(c-a) × x^c(a-b)

Since (a^m)^n = a^(mn)

=> x^(ab-ac) × x^(bc-ab) × x^(ac-bc)

=> x^(ab-ac+bc-ab+ac-bc)

Since a^m × a^n = a^(m+n)

=> x^(ab-ab+bc-bc+ac-ac)

=>x^0

=>1

Since a^0 = 1

=> RHS

Therefore, LHS = RHS

Answer:-

(x^b/x^c)^a(x^c/x^a)^b(x^a/x^b)^c = 1

Used formulae:-

• a^m/a^ = a^(m-n)
• (a^m)^n = a^(mn)
• a^m × a^n = a^(m+n)
• a^0 = 1