Express the following fraction in simplest form, only using positive exponents.
start fraction, 3, t, to the power minus 1 , divided by, minus, 3, left parenthesis, t, to the power minus 1 , right parenthesis, to the power 4 , end fraction
−3(t 
−1
 ) 
4
 
3t 
−1
 
​


Let's simplify the given expression:

\[
\frac{3t^{-1}}{-3(t^{-1})^4}
\]

### Step-by-Step Solution

1. **Simplify the denominator**:
   - The term \((t^{-1})^4\) means we need to raise \(t^{-1}\) to the power of 4.
   - Using the power of a power property \((a^m)^n = a^{m \cdot n}\):
     \[
     (t^{-1})^4 = t^{-4}
     \]
   - So, the denominator becomes:
     \[
     -3 \cdot t^{-4} = -3t^{-4}
     \]

   Now the fraction is:
   \[
   \frac{3t^{-1}}{-3t^{-4}}
   \]

2. **Divide the coefficients**:
   - Divide \(3\) by \(-3\):
     \[
     \frac{3}{-3} = -1
     \]

   So, the expression now becomes:
   \[
   -\frac{t^{-1}}{t^{-4}}
   \]

3. **Simplify the exponents**:
   - Use the property \(\frac{a^m}{a^n} = a^{m - n}\):
     \[
     \frac{t^{-1}}{t^{-4}} = t^{-1 - (-4)} = t^{3}
     \]

4. **Rewrite the result**:
   - Since we now have \(t^3\), the expression in its simplest form is:
     \[
     -t^3
     \]

### Final Answer
\[
-t^3
\]

Would you like further clarification on any step?

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Here are 5 related questions for further practice:

1. Simplify \(\frac{4x^{-2}}{-2x^{-3}}\).
2. Write \(\frac{y^3}{y^{-2}}\) in simplest form using only positive exponents.
3. Simplify \(\frac{6a^{-1}}{-2a^2}\).
4. Express \(\frac{-5b^{-3}}{10b^{-2}}\) in simplest form.
5. Simplify \(\frac{7c^{-2}}{-7c^{-5}}\).

**Tip:** When dividing terms with the same base, remember to subtract the exponents.

Express the following fraction in simplest form, only using positive exponents. \(\frac{3t^{-1}}{-3(t^{-1})^4}\)

Learn how to express \(\frac{3t^{-1}}{-3(t^{-1})^4}\) in its simplest form with positive exponents. This solution uses exponent rules and fraction simplification, suitable for high school algebra students in Grades 9-11.

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