We are given the equation:

\[
54 = 2 \left( -1 - 5t \right)^{3/2}
\]

Let's solve for \( t \) step by step.

### Step 1: Isolate the term with \( t \)
We start by dividing both sides of the equation by 2 to simplify:

\[
\frac{54}{2} = \left( -1 - 5t \right)^{3/2}
\]

\[
27 = \left( -1 - 5t \right)^{3/2}
\]

### Step 2: Eliminate the fractional exponent
To remove the exponent \( \frac{3}{2} \), we raise both sides of the equation to the power of \( \frac{2}{3} \):

\[
\left( 27 \right)^{\frac{2}{3}} = -1 - 5t
\]

Since \( 27^{\frac{2}{3}} = 9 \) (because \( 27^{1/3} = 3 \) and \( 3^2 = 9 \)), we get:

\[
9 = -1 - 5t
\]

### Step 3: Solve for \( t \)
Now we solve the linear equation:

\[
9 + 1 = -5t
\]

\[
10 = -5t
\]

Dividing both sides by -5:

\[
t = \frac{10}{-5} = -2
\]

### Final Answer:
\[
t = -2
\]

Would you like more details on any step or have any further questions?

Here are 5 related questions:
1. How do you deal with negative exponents in equations?
2. What is the process of simplifying fractional exponents?
3. How can you solve equations with radicals?
4. What are the steps to linearize equations involving powers?
5. Can fractional exponents be negative? How do they affect the solution?

**Tip**: Always check your solution by substituting the value of \( t \) back into the original equation to verify accuracy.

This page provides a detailed step-by-step solution for solving the equation 54 = 2(-1 - 5t)^(3/2), covering key algebraic concepts like fractional exponents and linear equations. Suitable for high school students in Grades 9-11.

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