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

To solve the equation $5 \cdot 5^{2x} = 125^{x+2}$, follow these steps:

1. **Rewrite the bases in terms of powers of 5**:
   Note that $125 = 5^3$, so we can rewrite the equation using this relationship:
   \[
   5 \cdot 5^{2x} = (5^3)^{x+2}
   \]

2. **Simplify the equation**:
   Using the property of exponents $(a^m)^n = a^{mn}$, the right-hand side becomes:
   \[
   5 \cdot 5^{2x} = 5^{3(x+2)}
   \]
   Simplify further:
   \[
   5 \cdot 5^{2x} = 5^{3x+6}
   \]

3. **Combine the exponents on the left side**:
   Recall that $a \cdot a^b = a^{1+b}$, so:
   \[
   5^{1 + 2x} = 5^{3x+6}
   \]

4. **Set the exponents equal to each other**:
   Since the bases are the same, the exponents must be equal:
   \[
   1 + 2x = 3x + 6
   \]

5. **Solve for $x$**:
   Isolate $x$ by moving terms involving $x$ to one side and constants to the other:
   \[
   1 + 2x - 3x = 6
   \]
   \[
   1 - x = 6
   \]
   \[
   -x = 5
   \]
   \[
   x = -5
   \]

Therefore, the solution to the equation is:
\[ x = -5 \]

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 equation 5 * 5^(2x) = 125^(x+2) with a detailed step-by-step solution. This problem involves exponential equations and properties of exponents. Suitable for advanced high school students.

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