Question

Using law of indices ​

Answer 1

1

Explanation:

There are a couple of identities that come into play here:

a³ -b³ = (a -b)(a² +ab +b²)

(x^a)/(x^b) = x^(a-b)

(x^a)^b = x^(ab)

(x^a)(x^b) = x^(a+b)

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These mean we can simplify the expression as follows.

`\left((x^(a))/(x^(b))\right)^(a^2+ab+b^2)*\left((x^(b))/(x^(c))\right)^(b^2+bc+c^2)*\left((x^(c))/(x^(a))\right)^(c^2+ca+a^2)\\\\=x^((a-b)(a^2+ab+b^2))* x^((b-c)(b^2+bc+c^2))* x^((c-a)(c^2+ca+a^2))\\\\=x^(a^3-b^3)* x^(b^3-c^3)* x^(c^3-a^3)=x^((a^3+b^3+c^3)-(a^3+b^3+c^3))\\\\=x^0=\boxed{1}`

The expression has a value of 1.

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Additional comment

For certain values of x and the exponents, the individual factors may exceed the ability of a calculator to express the value. That is, an attempt at numerical evaluation of this may produce a result different from 1. In any event, the expression is undefined for x=0.