find the value of x if (-3)^(2x)-:(-3)^(5)=(-3)^(7)

We start with the given equation:

\[
(-3)^{2x} \div (-3)^5 = (-3)^7
\]

### Step 1: Simplify the left side of the equation
Using the properties of exponents, specifically \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify the left side:

\[
\frac{(-3)^{2x}}{(-3)^5} = (-3)^{2x - 5}
\]

### Step 2: Set the exponents equal
The equation now becomes:

\[
(-3)^{2x - 5} = (-3)^7
\]

Since the bases are the same and non-zero, we can set the exponents equal to each other:

\[
2x - 5 = 7
\]

### Step 3: Solve for \(x\)
Add 5 to both sides:

\[
2x = 12
\]

Now, divide both sides by 2:

\[
x = 6
\]

So, the value of \(x\) is \(6\).

Would you like to explore the details of this solution, or do you have any additional questions?

Here are 5 related questions:
1. How would the solution change if the base was positive?
2. What happens if the exponent terms involve addition rather than subtraction?
3. Can this method be applied to bases other than integers?
4. How do you handle situations where the exponents are fractions?
5. What are some real-world applications of this kind of equation?

**Tip:** Always check the base in exponential equations; if it's the same across terms, you can directly equate the exponents.

Learn how to solve the exponential equation (-3)^(2x) / (-3)^5 = (-3)^7 step-by-step with properties of exponents. This problem involves simplifying exponents and setting them equal to each other. Suitable for Grades 9-12.

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