(-1/3)5 / (-3)10 = (-1/3)2n find value of n

We are given the equation:

\[
\frac{\left(\frac{-1}{3}\right)^5}{\left(-3\right)^{10}} = \left(\frac{-1}{3}\right)^{2n}
\]

### Step 1: Simplify the left-hand side of the equation.
We can rewrite the denominator \((-3)^{10}\) as:

\[
(-3)^{10} = \left(\frac{-1}{3}\right)^{10} \cdot 3^{10}
\]

So the left-hand side becomes:

\[
\frac{\left(\frac{-1}{3}\right)^5}{\left(\frac{-1}{3}\right)^{10}} = \left(\frac{-1}{3}\right)^{5 - 10} = \left(\frac{-1}{3}\right)^{-5}
\]

### Step 2: Rewrite the equation.

Now we have:

\[
\left(\frac{-1}{3}\right)^{-5} = \left(\frac{-1}{3}\right)^{2n}
\]

Since the bases are the same, we can set the exponents equal to each other:

\[
-5 = 2n
\]

### Step 3: Solve for \( n \).

\[
n = \frac{-5}{2}
\]

Thus, the value of \( n \) is:

\[
n = -\frac{5}{2}
\]

Would you like further clarification, or have any additional questions?

---

Here are 5 related questions for practice:

1. What is the value of \( n \) in \( \left(\frac{-1}{4}\right)^3 = \left(\frac{-1}{4}\right)^{n + 1} \)?
2. Simplify: \( \frac{\left(2/5\right)^6}{\left(2/5\right)^2} \).
3. Solve for \( x \): \( 3^x = 81 \).
4. How do you simplify \( (-2)^4 / (-2)^6 \)?
5. What is the value of \( x \) in \( \frac{(4^3)}{4^x} = 4 \)?

**Tip:** When dealing with exponents, remember that dividing powers of the same base means subtracting the exponents.

(-1/3)^5 / (-3)^10 = (-1/3)^{2n} find value of n

Discover how to solve the exponential equation (-1/3)^5 / (-3)^10 = (-1/3)^{2n} and find the value of n. This step-by-step solution covers the laws of exponents, base equality, and simplification techniques. Ideal for students in Grades 9-11.

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