%5Ea%2Bb*(x%5Eb%2Fx%5Ec)%5Eb%2Bc*(x%5Ec%2Fx%5Ea)%5Ec%2Ba.jpeg "(x^a/x^b)^a+b*(x^b/x^c)^b+c*(x^c/x^a)^c+a")
Simplifying the Expression (x^a/x^b)^a+b*(x^b/x^c)^b+c*(x^c/x^a)^c+a
This expression looks complex, but we can simplify it using the properties of exponents. Let's break it down step-by-step.
Applying the Quotient Rule of Exponents
The quotient rule states that when dividing exponents with the same base, you subtract the powers: x^m / x^n = x^(m-n)
Applying this rule to each term in our expression:
- (x^a/x^b)^a+b = (x^(a-b))^(a+b)
- (x^b/x^c)^b+c = (x^(b-c))^(b+c)
- (x^c/x^a)^c+a = (x^(c-a))^(c+a)
Applying the Power Rule of Exponents
The power rule states that when raising a power to another power, you multiply the exponents: (x^m)^n = x^(m*n)
Applying this rule to each term:
- (x^(a-b))^(a+b) = x^((a-b)*(a+b)) = x^(a^2 - b^2)
- (x^(b-c))^(b+c) = x^((b-c)*(b+c)) = x^(b^2 - c^2)
- (x^(c-a))^(c+a) = x^((c-a)*(c+a)) = x^(c^2 - a^2)
Combining the Simplified Terms
Now our expression becomes:
x^(a^2 - b^2) * x^(b^2 - c^2) * x^(c^2 - a^2)
Using the product rule of exponents (which states x^m * x^n = x^(m+n)), we can combine the terms:
x^(a^2 - b^2 + b^2 - c^2 + c^2 - a^2)
Simplifying further, we notice that all the terms cancel out:
x^0
The Final Answer
Finally, any number raised to the power of 0 equals 1. Therefore:
(x^a/x^b)^a+b*(x^b/x^c)^b+c*(x^c/x^a)^c+a = 1