To integrate the expression

\[
\int \frac{dt}{t^{4/3}}
\]

we can rewrite the integrand in a simpler form. Notice that:

\[
\frac{1}{t^{4/3}} = t^{-4/3}
\]

So the integral becomes:

\[
\int t^{-4/3} \, dt
\]

### Step 1: Apply the Power Rule
Using the power rule for integration, which states that \(\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\) (where \(n \neq -1\)), we can integrate \(t^{-4/3}\).

Here, \(n = -\frac{4}{3}\), so:

\[
\int t^{-4/3} \, dt = \frac{t^{-4/3 + 1}}{-4/3 + 1} + C
\]

### Step 2: Simplify the Exponent
Calculate \(-\frac{4}{3} + 1\):

\[
-\frac{4}{3} + 1 = -\frac{4}{3} + \frac{3}{3} = -\frac{1}{3}
\]

Thus, we have:

\[
\int t^{-4/3} \, dt = \frac{t^{-1/3}}{-1/3} + C
\]

### Step 3: Simplify the Fraction
Dividing by \(-\frac{1}{3}\) is equivalent to multiplying by \(-3\):

\[
= -3t^{-1/3} + C
\]

### Step 4: Rewrite in Terms of Positive Exponents
Finally, we can write \(t^{-1/3}\) as \(\frac{1}{t^{1/3}}\):

\[
\int \frac{dt}{t^{4/3}} = -\frac{3}{t^{1/3}} + C
\]

### Final Answer
\[
\int \frac{dt}{t^{4/3}} = -\frac{3}{t^{1/3}} + C
\]

Would you like a detailed explanation of each step, or do you have any questions?

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Here are 5 related questions to deepen understanding:
1. How would you evaluate \(\int \frac{dt}{t^{5/3}}\)?
2. What is the general formula for integrating expressions of the form \(\int t^n \, dt\)?
3. How does the result change if we integrate \(\int \frac{dt}{t^{2}}\) instead?
4. What is the power rule, and why does it not apply to \(\int \frac{1}{t} \, dt\)?
5. How can we apply similar techniques to evaluate \(\int \frac{1}{t^{a}} \, dt\) for any constant \(a\)?

**Tip:** When working with fractional exponents, rewrite terms to apply integration rules more easily.

Learn how to solve the integral \( \int \frac{dt}{t^{4/3}} \) using the power rule. This detailed step-by-step guide simplifies fractional exponents and explains each step, from rewriting the integrand to applying the power rule and simplifying the result. Ideal for high school students.

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