`(frac(-1)(5))^(3) div (frac(-1)(5))^(8)=` ?

Options

A. (a) `(-frac(1)(5))^(5)`
B. (b) `(frac(-1)(5))^(11)`
C. (c) `(-5)^(5)`
D. (d) `(frac(1)(5))^(5)`

Answer

To solve the expression \((\frac{-1}{5})^{3} \div (\frac{-1}{5})^{8}\), we can use the laws of exponents. Here’s a step-by-step solution:

### Step 1: Identify the bases and apply the division rule for exponents
The bases in both parts of the expression are the same, which is \(-\frac{1}{5}\). According to the laws of exponents, when dividing two expressions with the same base, we subtract the exponents.

\[ \frac{(-1/5)^{3}}{(-1/5)^{8}} = (-1/5)^{3-8} \]

### Step 2: Simplify the exponent
Now, we simplify the exponent:

\[ 3 - 8 = -5 \]

So, we have:

\[ (-1/5)^{-5} \]

### Step 3: Apply the negative exponent rule
A negative exponent means we take the reciprocal of the base. Therefore:

\[ (-1/5)^{-5} = \frac{1}{(-1/5)^{5}} \]

### Step 4: Calculate \((-1/5)^{5}\)
Now we need to calculate \((-1/5)^{5}\):

\[ (-1/5)^{5} = \frac{(-1)^{5}}{5^{5}} = \frac{-1}{5^{5}} \]

### Step 5: Calculate \(5^{5}\)
Now, we calculate \(5^{5}\):

\[ 5^{5} = 5 \times 5 \times 5 \times 5 \times 5 = 3125 \]

### Step 6: Substitute back into the expression
Now we substitute back into our expression:

\[ (-1/5)^{-5} = \frac{1}{\frac{-1}{3125}} = -3125 \]

### Final Answer
Thus, the final answer is:

\[ (\frac{-1}{5})^{3} \div (\frac{-1}{5})^{8} = -3125 \]

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