# For Any Positive Real Number X, Find the Value of - Mathematics

For any positive real number x, find the value of $\left( \frac{x^a}{x^b} \right)^{a + b} \times \left( \frac{x^b}{x^c} \right)^{b + c} \times \left( \frac{x^c}{x^a} \right)^{c + a}$.

#### Solution

We have to find the value of =   (x^a/x^a)^(a+b) xx (x^b/x^c)^(b+c) xx(x^c/x^a)^(c+a)

L=(x^(a(a+b))/(x^(b(a+b)))) xx (x^(b(b+c))/(x^(c(b+c)))) xx  (x^(c(c+b))/(x^(a(c+a))))

= (x^(a^2+ab))/ (x^(ba+b^2)) xx (x^(b^2+bc))/ (x^(bc+c^2)) xx (x^(c^2+ca))/ (x^(ac+b^2))

By using rational exponents, a^mxx a^n = a^(m+n) we get

L=(x^(a^2+ab+b^2+bc+c^2+ca))/(x^(ab+b^2+bc+c^2+ac+a^2))

By using rational exponents    a^m/a^n= a^(m-n) we get

L = x^((a^2+ab+b^2+bc+c^2+ca) -(ab+b^2+bc+c^2+ac+a^2))

=x^((a^2+ab+b^2+bc+c^2+ca) -(ab+b^2+bc+c^2+ac+a^2))

=x^0

By definition we can write x^0 as 1

Hence the value of expression is 1.

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 2 Exponents of Real Numbers
Q 10 | Page 29