`(2d^(2)e^(-1))^(3)xx(d^(3)/e)^(-2)=`

Options

A. `8e^(-2)`
B. `8e^(-3)`
C. `8e^(-1)`
D. `8e^(-4)`

Answer

To solve the expression \((2d^{2}e^{-1})^{3} \times (d^{3}/e)^{-2}\), we will follow the properties of exponents step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ (2d^{2}e^{-1})^{3} \times (d^{3}/e)^{-2} \]

### Step 2: Apply the power of a product rule
Using the property \((a \cdot b)^{n} = a^{n} \cdot b^{n}\), we can rewrite the first part:
\[ = 2^{3} \cdot (d^{2})^{3} \cdot (e^{-1})^{3} \times (d^{3})^{-2} \cdot (e^{-1})^{-2} \]

### Step 3: Calculate the individual powers
Now we calculate each part:
- \(2^{3} = 8\)
- \((d^{2})^{3} = d^{6}\) (using \((a^{m})^{n} = a^{m \cdot n}\))
- \((e^{-1})^{3} = e^{-3}\)
- \((d^{3})^{-2} = d^{-6}\)
- \((e^{-1})^{-2} = e^{2}\)

So we can rewrite the expression as:
\[ = 8 \cdot d^{6} \cdot e^{-3} \cdot d^{-6} \cdot e^{2} \]

### Step 4: Combine like terms
Now we combine the terms with the same base:
- For \(d\): \(d^{6} \cdot d^{-6} = d^{6 - 6} = d^{0} = 1\)
- For \(e\): \(e^{-3} \cdot e^{2} = e^{-3 + 2} = e^{-1}\)

Thus, the expression simplifies to:
\[ = 8 \cdot 1 \cdot e^{-1} = 8e^{-1} \]

### Final Result
The final result of the expression is:
\[ 8e^{-1} \]