Simplify `((1)/(2)) ^(-2) + ((1)/(3)) ^(-2) + ((1)/(4)) ^(-2)`

Options

A. `29`
B. `39`
C. `49`
D. `19`

Answer

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

### 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:

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

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

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

### Step 2: Substitute Back into the Expression
Now we can substitute these values back into the original expression:

\[ 4 + 9 + 16 \]

### Step 3: Perform the Addition
Now, we simply add the numbers together:

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

### Final Answer
Thus, the simplified expression is:

\[ \boxed{29} \]

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