5x5 $$ ^{\left(2x\right)} $$=125$$ ^{\left(x+2\right)} $$

To solve the equation $5^{5(2x)} = 125^{(x+2)}$, we first need to express the bases as powers of a common base.

Note that:
\[ 125 = 5^3 \]

Rewriting the equation, we get:
\[ 5^{5(2x)} = (5^3)^{(x+2)} \]

Next, we use the property of exponents $(a^m)^n = a^{mn}$:
\[ 5^{10x} = 5^{3(x+2)} \]

Since the bases are the same, we can set the exponents equal to each other:
\[ 10x = 3(x + 2) \]

Now, solve for $x$:
\[ 10x = 3x + 6 \]
\[ 10x - 3x = 6 \]
\[ 7x = 6 \]
\[ x = \frac{6}{7} \]

Therefore, the solution to the equation is:
\[ x = \frac{6}{7} \]

Would you like more details or have any questions?

Here are 5 questions you can ask next:
1. How do you solve equations involving logarithms?
2. What is the quadratic formula and how is it derived?
3. Can you explain the properties of exponents?
4. How do you solve systems of linear equations using matrices?
5. What is the difference between permutations and combinations?

**Tip:** When solving exponential equations, always try to express all terms with the same base if possible. This simplifies the equation and makes it easier to solve.

Learn how to solve the exponential equation 5^(5(2x)) = 125^(x+2) with a step-by-step solution. This problem involves applying properties of exponents and converting bases to solve for the variable x. Suitable for students in Grades 10-12.

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