If x = y 1/a , y = z 1/b and z = x 1/c where x ≠ 1, y ≠ 1 and z ≠ 1, then what is the value of abc?

Question

If x = y 1/a , y = z 1/b and z = x 1/c where x &ne; 1, y &ne; 1 and z &ne; 1, then what is the value of abc?

Prepp Team · Accepted Answer

Finding the Value of abc in Exponential Equations

This problem involves solving a system of equations where variables are related through exponents. We are given three equations:

\(x = y^{1/a}\)
\(y = z^{1/b}\)
\(z = x^{1/c}\)

We are also told that \(x \ne 1\), \(y \ne 1\), and \(z \ne 1\). Our goal is to find the value of the product \(abc\).

Let's substitute the expressions from one equation into another to find a relationship involving all three variables and the exponents \(a\), \(b\), and \(c\). We can start by substituting the expression for \(y\) from equation (2) into equation (1):

From equation (1): \(x = y^{1/a}\)
Substitute \(y = z^{1/b}\) into this equation:
\(x = (z^{1/b})^{1/a}\)

Using the rule of exponents \((p^m)^n = p^{mn}\), we can simplify the right side:

\(x = z^{(1/b) \times (1/a)}\)
\(x = z^{1/(ab)}\)

Now we have a relationship between \(x\) and \(z\) involving \(a\) and \(b\). Next, let's substitute the expression for \(z\) from equation (3) into this new equation:

We have: \(x = z^{1/(ab)}\)
Substitute \(z = x^{1/c}\) into this equation:
\(x = (x^{1/c})^{1/(ab)}\)

Again, using the rule of exponents \((p^m)^n = p^{mn}\), we simplify the right side:

\(x = x^{(1/c) \times (1/(ab))}\)
\(x = x^{1/(abc)}\)

So we have the equation \(x = x^{1/(abc)}\).

We are given that \(x \ne 1\). When we have an equation of the form \(X = X^P\) where \(X \ne 1\) (and \(X \ne 0\)), this equation holds true only if the exponent \(P\) is equal to 1.

In our equation \(x = x^{1/(abc)}\), the base is \(x\) and the exponent is \(1/(abc)\). Since \(x \ne 1\), for this equation to be true, the exponent must be 1.

Therefore, we must have:

\( \frac{1}{abc} = 1 \)

To find the value of \(abc\), we can take the reciprocal of both sides of this equation:

\( abc = \frac{1}{1} \) \( abc = 1 \)

The value of \(abc\) is 1. The conditions \(y \ne 1\) and \(z \ne 1\) are consistent with this result, as they prevent scenarios where the bases are 1, which would make the equations true for exponents other than 1.

Thus, based on the given relationships and conditions, the value of \(abc\) is 1.

Revision Table: Key Concepts

Concept	Description	Relevance to Problem
Exponent Rule: \((p^m)^n = p^{mn}\)	When raising a power to another power, multiply the exponents.	Used to simplify \((z^{1/b})^{1/a}\) and \((x^{1/c})^{1/(ab)}\).
Equation \(X = X^P\)	If the base \(X \ne 1\) (and \(X \ne 0\)), the equation holds if and only if \(P=1\).	Used to deduce \(1/(abc) = 1\) from \(x = x^{1/(abc)}\).
Given Conditions (\(x, y, z \ne 1\))	These conditions are crucial as they allow us to equate exponents when bases are equal.	Ensures that we can conclude \(1/(abc)=1\) from \(x=x^{1/(abc)}\).

Additional Information: Working with Exponential Equations

When dealing with exponential equations, several rules of exponents are fundamental. Besides \((p^m)^n = p^{mn}\), other important rules include:

Product Rule: \(p^m \times p^n = p^{m+n}\) (When multiplying powers with the same base, add the exponents).
Quotient Rule: \(\frac{p^m}{p^n} = p^{m-n}\) (When dividing powers with the same base, subtract the exponents).
Zero Exponent Rule: \(p^0 = 1\) (Any non-zero number raised to the power of 0 is 1).
Negative Exponent Rule: \(p^{-n} = \frac{1}{p^n}\) (A base raised to a negative exponent is equal to the reciprocal of the base raised to the positive exponent).

In this problem, we used substitution repeatedly to consolidate the relationships into a single equation involving only one variable (\(x\)) and the product of the exponents (\(abc\)). This is a common strategy for solving systems of equations, including those with exponents.