`(-27a^(10))^(3/5)`
For all values of a, which of the following is equivalent to the expression above?

Options

A. `3a^6root5 3`
B. `-3a^6root5 81`
C. `3a^5root5 81`
D. `-3a^5root5 3`

Answer

To solve the expression \((-27a^{10})^{\frac{3}{5}}\), we will follow the laws of exponents step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ (-27a^{10})^{\frac{3}{5}} \]
We can separate the components inside the parentheses:
\[ (-27)^{\frac{3}{5}} \cdot (a^{10})^{\frac{3}{5}} \]

### Step 2: Simplify \((-27)^{\frac{3}{5}}\)
Next, we simplify \((-27)^{\frac{3}{5}}\). We can express \(-27\) as \(-3^3\):
\[ (-3^3)^{\frac{3}{5}} = (-1)^{\frac{3}{5}} \cdot (3^3)^{\frac{3}{5}} = (-1)^{\frac{3}{5}} \cdot 3^{\frac{9}{5}} \]
Since \((-1)^{\frac{3}{5}}\) is a complex number, we will focus on the real part for simplification.

### Step 3: Simplify \( (a^{10})^{\frac{3}{5}} \)
Now we simplify \( (a^{10})^{\frac{3}{5}} \):
\[ (a^{10})^{\frac{3}{5}} = a^{10 \cdot \frac{3}{5}} = a^6 \]

### Step 4: Combine the results
Now we combine the results from Steps 2 and 3:
\[ (-27a^{10})^{\frac{3}{5}} = (-1)^{\frac{3}{5}} \cdot 3^{\frac{9}{5}} \cdot a^6 \]

### Step 5: Rewrite \(3^{\frac{9}{5}}\)
We can rewrite \(3^{\frac{9}{5}}\) as:
\[ 3^{\frac{9}{5}} = 3^{1 + \frac{4}{5}} = 3^1 \cdot 3^{\frac{4}{5}} = 3 \cdot 3^{\frac{4}{5}} \]

### Final Expression
Thus, the final expression becomes:
\[ (-1)^{\frac{3}{5}} \cdot 3 \cdot 3^{\frac{4}{5}} \cdot a^6 \]
This can be represented as:
\[ -3 \cdot a^6 \cdot 3^{\frac{4}{5}} \quad \text{(considering the real part)} \]

### Conclusion
The expression \((-27a^{10})^{\frac{3}{5}}\) simplifies to:
\[ -3 \cdot a^6 \cdot 3^{\frac{4}{5}} \]
This can also be expressed as:
\[ -3 \cdot a^6 \cdot \sqrt[5]{81} \]