`{(frac(3)(4))^(-1)-(frac(1)(4))^(-1)}^(-1)=` ?

Options

A. (a) `frac(3)(8)`
B. (b) `frac(-3)(8)`
C. (c) `frac(8)(3)`
D. (d) `frac(-8)(3)`

Answer

To solve the equation \(\left(\left(\frac{3}{4}\right)^{-1} - \left(\frac{1}{4}\right)^{-1}\right)^{-1}\), we will follow these steps:

### Step 1: Simplify the individual terms with negative exponents
Using the property of exponents that \(a^{-n} = \frac{1}{a^n}\), we can rewrite the terms:
\[ \left(\frac{3}{4}\right)^{-1} = \frac{4}{3} \]
\[ \left(\frac{1}{4}\right)^{-1} = 4 \]

### Step 2: Substitute back into the expression
Now we substitute these values back into the original expression:
\[ \left(\frac{4}{3} - 4\right)^{-1} \]

### Step 3: Find a common denominator and simplify
To subtract \(\frac{4}{3}\) and \(4\), we need a common denominator. The common denominator is \(3\):
\[ 4 = \frac{12}{3} \]
Now we can perform the subtraction:
\[ \frac{4}{3} - \frac{12}{3} = \frac{4 - 12}{3} = \frac{-8}{3} \]

### Step 4: Take the reciprocal of the result
Now we take the reciprocal of the result:
\[ \left(\frac{-8}{3}\right)^{-1} = \frac{3}{-8} = -\frac{3}{8} \]

### Final Answer
Thus, the final answer is:
\[ -\frac{3}{8} \]

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