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Prove That: `(X^A/X^B)^(A^2+Ab+B^2)Xx(X^B/X^C)^(B^2+Bc+C^2)Xx(X^C/X^A)^(C^2+Ca+A^2)=1` - Mathematics

Prove that:

`(x^a/x^b)^(a^2+ab+b^2)xx(x^b/x^c)^(b^2+bc+c^2)xx(x^c/x^a)^(c^2+ca+a^2)=1`

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Solution

Consider the left hand side:

`(x^a/x^b)^(a^2+ab+b^2)xx(x^b/x^c)^(b^2+bc+c^2)xx(x^c/x^a)^(c^2+ca+a^2)`

`=x^(a(a^2+ab+b^2))/x^(b(a^2+ab+b^2))xxx^(b(b^2+bc+c^2))/x^(c(b^2+bc+c^2))xxx^(c(c^2+ca+a^2))/x^(a(c^2+ca+a^2))`

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

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

`=x^((a^3-b^3))xxx((b^3-c^3))xxx^((c^3-a^3))`

`=x^((a^3-b^3+b^3-c^3+c^3-a^3))`

`=x^0`

= 1

Left hand side is equal to right hand side.
Hence proved.

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APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 2 Exponents of Real Numbers
Exercise 2.1 | Q 3.1 | Page 12
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