Simplify: `(2^(-1)div5^(-1))^(2) xx (frac(-5)(8))^(-1)`

Options

A. 10
B. -10
C. 20
D. -20

Answer

To simplify the expression \((2^{-1} \div 5^{-1})^{2} \times \left(-\frac{5}{8}\right)^{-1}\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (2^{-1} \div 5^{-1})^{2} \times \left(-\frac{5}{8}\right)^{-1} \]

### Step 2: Change negative exponents to positive
Using the property of exponents, \(a^{-n} = \frac{1}{a^n}\), we can rewrite the negative exponents:
\[ (2^{-1} \div 5^{-1})^{2} = \left(\frac{1}{2} \div \frac{1}{5}\right)^{2} = \left(\frac{1}{2} \times 5\right)^{2} \]
And for \(\left(-\frac{5}{8}\right)^{-1}\):
\[ \left(-\frac{5}{8}\right)^{-1} = -\frac{8}{5} \]

### Step 3: Combine the expressions
Now, substituting back into the expression:
\[ \left(\frac{5}{2}\right)^{2} \times \left(-\frac{8}{5}\right) \]

### Step 4: Calculate \(\left(\frac{5}{2}\right)^{2}\)
Calculating the square:
\[ \left(\frac{5}{2}\right)^{2} = \frac{5^{2}}{2^{2}} = \frac{25}{4} \]

### Step 5: Multiply by \(-\frac{8}{5}\)
Now we multiply:
\[ \frac{25}{4} \times \left(-\frac{8}{5}\right) = -\frac{25 \times 8}{4 \times 5} \]

### Step 6: Simplify the multiplication
Calculating the numerator and denominator:
\[ -\frac{200}{20} = -10 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{-10} \]