https://file.aiquickdraw.com/mathbot/file/172815206117436r1scsf.jpg

The equation given in the image is:

\[
\left(\frac{x^m}{x^n}\right)^t \cdot \left(\frac{x^n}{x^t}\right)^m \cdot \left(\frac{x^t}{x^m}\right)^n = 1
\]

To prove this, we need to simplify each part using the properties of exponents.

### Step-by-step solution:

1. **Simplify the first term**: 
   \[
   \left(\frac{x^m}{x^n}\right)^t = \left(x^{m-n}\right)^t = x^{t(m-n)}
   \]

2. **Simplify the second term**: 
   \[
   \left(\frac{x^n}{x^t}\right)^m = \left(x^{n-t}\right)^m = x^{m(n-t)}
   \]

3. **Simplify the third term**:
   \[
   \left(\frac{x^t}{x^m}\right)^n = \left(x^{t-m}\right)^n = x^{n(t-m)}
   \]

4. **Multiply all the terms together**:
   \[
   x^{t(m-n)} \cdot x^{m(n-t)} \cdot x^{n(t-m)}
   \]
   Using the exponent rule \(a^p \cdot a^q = a^{p+q}\), we can combine the exponents:
   \[
   x^{t(m-n) + m(n-t) + n(t-m)}
   \]

5. **Simplify the exponent**:
   \[
   t(m-n) + m(n-t) + n(t-m)
   \]
   Expanding this expression:
   \[
   tm - tn + mn - mt + nt - nm
   \]
   Notice that all terms cancel out:
   \[
   tm - tn + mn - mt + nt - nm = 0
   \]

6. **Conclusion**:
   The exponent simplifies to 0, so:
   \[
   x^0 = 1
   \]

Thus, the given expression simplifies to 1, as required.

---

Do you want more details on any of these steps? 

Here are 5 follow-up questions you might be interested in:
1. What are some common rules of exponents used in such proofs?
2. How do you simplify complex algebraic fractions involving exponents?
3. Could this problem be generalized for different values of \(m\), \(n\), and \(t\)?
4. How do we deal with negative exponents in similar problems?
5. What happens if the base \(x\) is negative or a fraction?

**Tip:** Always remember that \(x^0 = 1\) for any non-zero value of \(x\).

Prove that (x^m / x^n)^t * (x^n / x^t)^m * (x^t / x^m)^n = 1.

In this tutorial, learn how to prove that (x^m / x^n)^t * (x^n / x^t)^m * (x^t / x^m)^n equals 1. The solution involves applying key exponent rules such as division of exponents and multiplication of like bases. Suitable for high school students, Grades 9-12.

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