Answer:

To simplify the expression 1/(x^b x^-c 1) * 1/(x^c x^-a 1) * 1/(x^a x^-b 1), we can use the properties of exponents and the rules of multiplication and division of fractions.

Step 1: Simplify the exponents

Let's start by simplifying the exponents in each term separately.

In the first term, we have x^b x^-c. According to the rule of exponents, when we multiply two terms with the same base, we add their exponents. So, x^b x^-c can be written as x^(b + (-c)), which simplifies to x^(b - c).

In the second term, we have x^c x^-a. Again, using the rule of exponents, we add the exponents to get x^(c + (-a)), which simplifies to x^(c - a).

In the third term, we have x^a x^-b. Using the same rule of exponents, we add the exponents to get x^(a + (-b)), which simplifies to x^(a - b).

So, the expression simplifies to 1/(x^(b - c) * x^(c - a) * x^(a - b) * 1).

Step 2: Combine the terms

Now, let's combine the terms in the numerator and denominator.

In the numerator, we have 1 * 1 * 1 = 1.

In the denominator, we have x^(b - c) * x^(c - a) * x^(a - b) * 1. According to the rule of exponents, when we divide two terms with the same base, we subtract their exponents. So, x^(b - c) / x^(c - a) can be written as x^((b - c) - (c - a)), which simplifies to x^(b - c - c + a), further simplifying to x^(a - 2c + a).

Similarly, x^(a - 2c + a) / x^(a - b) can be written as x^((a - 2c + a) - (a - b)), which simplifies to x^(a - 2c + a - a + b), further simplifying to x^(b - 2c + b).

So, the expression becomes 1 / (x^(b - 2c + b) * 1).

Step 3: Simplify the expression further

Finally, we have 1 / (x^(b - 2c + b) * 1). Any term multiplied by 1 remains the same, so we can remove the 1. Also, 1 divided by a term is the same as that term raised to the power of -1.

Therefore, the expression simplifies to x^(-b + 2c - b).

Step 4: Determine the value of a, b, and c

Now, we need to determine the value of a, b, and c that make the expression x^(-b + 2c - b) equal to a single value.

Since x^(-b + 2c - b