The value of $\left(\frac{1}{2} ight)^{-2} imes \left(\frac{1}{3} ight)^{-2} imes \left(\frac{1}{4} ight)^{-2}$ is

Question

The value of $\left(\frac{1}{2}ight)^{-2} 	imes \left(\frac{1}{3}ight)^{-2} 	imes \left(\frac{1}{4}ight)^{-2}$ is

Accepted Answer

To solve the problem and verify why Option 2 is the correct answer, we need to evaluate the expression $\left(\frac{1}{2}\right)^{-2} \times \left(\frac{1}{3}\right)^{-2} \times \left(\frac{1}{4}\right)^{-2}$.

### Step-by-step Explanation:

1. **Understanding Negative Exponents:**
   - A negative exponent indicates the reciprocal of the base raised to the positive of that exponent. Therefore, $\left(\frac{1}{a}\right)^{-b} = a^b$.

2. **Applying the Rule to Each Term:**
   - $\left(\frac{1}{2}\right)^{-2} = 2^2 = 4$
   - $\left(\frac{1}{3}\right)^{-2} = 3^2 = 9$
   - $\left(\frac{1}{4}\right)^{-2} = 4^2 = 16$

3. **Multiplying the Results:**
   - Now, multiply the results of each term: $4 \times 9 \times 16$.

4. **Calculating the Product:**
   - First, calculate $4 \times 9 = 36$.
   - Then, multiply the result by 16: $36 \times 16 = 576$.

Thus, the value of the expression is 576, which confirms that Option 2 is indeed the correct answer.

### Explanation of Incorrect Options:

- **Option 1: -576**
  - The calculation involves only positive numbers, as all bases are positive and the exponents are even, resulting in positive products. Therefore, a negative result is not possible.

- **Option 3: 144**
  - This value is incorrect because the multiplication of the terms $4 \times 9 \times 16$ results in 576, not 144.

- **Option 4: 288**
  - Similarly, this value is incorrect because the correct multiplication of the terms yields 576, not 288.

In conclusion, the correct answer is Option 2, which is 576, as demonstrated by the calculations above.