$x^{x^7}=5^{(25)^{0.2}}$

Key takeaway: This equation can be handled cleanly by rewriting both sides as powers of 5 and matching exponents.

Step 1: Rewrite the right-hand side

author? Wait: keep the logic direct.

$25^{0.2}=25^{1/5}=(5^2)^{1/5}=5^{2/5}.$

So the equation becomes

$x^{x^7}=5^{5^{2/5}}.$

Step 2: Try a power-of-5 form

Let

$x=5^{1/5}.$

Then

$x^7=\left(5^{1/5}\right)^7=5^{7/5}.$

Now compute the left-hand side:

$x^{x^7}=(5^{1/5})^{5^{7/5}}=5^{\frac{1}{5}\cdot 5^{7/5}}=5^{5^{2/5}}.$

This matches the right-hand side exactly.

Answer

$\boxed{x=5^{1/5}}$

Why this works

The key is to recognize that the exponent on the right is already a power of 5:

$25^{0.2}=5^{2/5}.$

Choosing $x=5^{1/5}$ makes the exponent on the left become

$x^7=5^{7/5},$

so the exponentiation rule gives the same nested power structure.

---

Pitfalls the pros know 👇 A common mistake is to treat $25^{0.2}$ as $5^{0.2}$ or to simplify $x^{x^7}$ by separating the exponent incorrectly. For expressions like $x^{x^7}$, the base and exponent are linked, so you must substitute carefully and keep the exponentiation order intact.

What if the problem changes? If the equation were $x^{x^n}=5^{(25)^{0.2}}$ for some other $n$, the same method would still work only if you choose $x$ so that $x^n$ becomes a convenient power of 5. In this problem, $n=7$ fits perfectly with $x=5^{1/5}$ because it produces the matching exponent $5^{2/5}$.

Tags: exponential equation, power of a power, logarithmic matching