`((1)/(2))^(-2) + ((1)/(3))^(-2) + ((1 )/(4))^(-2)` = ?

Options

A. `(61)/(144)`
B. `(144)/(61)`
C. `29`
D. `1/29`

Answer

To solve the expression \(\left(\frac{1}{2}\right)^{-2} + \left(\frac{1}{3}\right)^{-2} + \left(\frac{1}{4}\right)^{-2}\), we will follow the steps below:

### Step 1: Apply the Negative Exponent Rule
The negative exponent rule states that \(a^{-n} = \frac{1}{a^n}\). Therefore, we can rewrite each term in the expression:

\[ \left(\frac{1}{2}\right)^{-2} = \frac{1}{\left(\frac{1}{2}\right)^{2}} = \left(\frac{2}{1}\right)^{2} = 2^{2} = 4 \]

\[ \left(\frac{1}{3}\right)^{-2} = \frac{1}{\left(\frac{1}{3}\right)^{2}} = \left(\frac{3}{1}\right)^{2} = 3^{2} = 9 \]

\[ \left(\frac{1}{4}\right)^{-2} = \frac{1}{\left(\frac{1}{4}\right)^{2}} = \left(\frac{4}{1}\right)^{2} = 4^{2} = 16 \]

### Step 2: Add the Results
Now that we have simplified each term, we can add them together:

\[ 4 + 9 + 16 \]

### Step 3: Calculate the Sum
Now, we will calculate the sum:

\[ 4 + 9 = 13 \]
\[ 13 + 16 = 29 \]

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

\[ \left(\frac{1}{2}\right)^{-2} + \left(\frac{1}{3}\right)^{-2} + \left(\frac{1}{4}\right)^{-2} = 29 \]