https://file.aiquickdraw.com/mathbot/file/1727455049834a3n7lzro.jpg

The equation from the image is:

\[
\left( \frac{-2}{5} \right)^7 = \left( \frac{-5}{2} \right)^x
\]

### Solving for \( x \):

1. Start by rewriting both sides of the equation using exponent rules.

\[
\left( \frac{-5}{2} \right) = \left( \frac{-2}{5} \right)^{-1}
\]

This is because flipping a fraction is equivalent to raising it to the power of \(-1\).

2. Therefore, the equation becomes:

\[
\left( \frac{-2}{5} \right)^7 = \left( \frac{-2}{5} \right)^{-x}
\]

3. Since the bases are the same, we can equate the exponents:

\[
7 = -x
\]

4. Solving for \( x \):

\[
x = -7
\]

Thus, the solution is \( x = -7 \).

Would you like more details or have any other questions?

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Here are 5 related questions to expand your understanding:

1. What are the properties of negative exponents and how do they affect fraction flipping?
2. How would the solution change if the base fractions were positive instead of negative?
3. Can this problem be solved using logarithms, and if so, how?
4. What is the significance of matching bases when solving exponential equations?
5. How can exponent rules be applied to more complex equations involving fractions?

**Tip:** Whenever you see a fraction raised to a negative exponent, remember that it means to take the reciprocal of the fraction.

Solve the equation: ((-2)/5)^7 = ((-5)/2)^x

Learn how to solve the exponential equation ((-2)/5)^7 = ((-5)/2)^x. This solution explores properties of exponents, reciprocal fractions, and matching bases. Suitable for students in Grades 9-11.

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