Understanding the Problem: Evaluating Exponents

The question asks us to find the value of a mathematical expression involving fractions raised to negative exponents. The expression is \( \left(\frac{1}{2}\right)^{−2} \times\left(\frac{1}{3}\right)^{−2} \times\left(\frac{1}{4}\right)^{−2} \). To solve this, we need to understand how to handle negative exponents, especially with fractions, and then perform the multiplication.

Rules for Negative Exponents

A key rule when evaluating exponents, particularly negative ones, is \(a^{-n} = \frac{1}{a^n}\) for any non-zero number \(a\). When dealing with a fraction \(\left(\frac{a}{b}\right)^{-n}\), the rule becomes even simpler: you can flip the fraction and change the sign of the exponent. That is,

\( \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n} \)

Using this rule helps significantly in simplifying expressions with negative exponents.

Step-by-Step Evaluation

We will apply the rule \(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}\) to each term in the mathematical expression:

1. First term: \( \left(\frac{1}{2}\right)^{−2} \)
2. Second term: \( \left(\frac{1}{3}\right)^{−2} \)
3. Third term: \( \left(\frac{1}{4}\right)^{−2} \)

Evaluating Each Term Separately

• For the first term, \( \left(\frac{1}{2}\right)^{−2} \):
  Using the rule, we flip the fraction \(\frac{1}{2}\) to \(\frac{2}{1}\) (which is just 2) and change the exponent to positive 2. So, \( \left(\frac{1}{2}\right)^{−2} = \left(\frac{2}{1}\right)^{2} = 2^2 \).
  \( 2^2 = 2 \times 2 = 4 \).
• For the second term, \( \left(\frac{1}{3}\right)^{−2} \):
  Similarly, we flip the fraction \(\frac{1}{3}\) to \(\frac{3}{1}\) (which is 3) and change the exponent to positive 2. So, \( \left(\frac{1}{3}\right)^{−2} = \left(\frac{3}{1}\right)^{2} = 3^2 \).
  \( 3^2 = 3 \times 3 = 9 \).
• For the third term, \( \left(\frac{1}{4}\right)^{−2} \):
  We flip the fraction \(\frac{1}{4}\) to \(\frac{4}{1}\) (which is 4) and change the exponent to positive 2. So, \( \left(\frac{1}{4}\right)^{−2} = \left(\frac{4}{1}\right)^{2} = 4^2 \).
  \( 4^2 = 4 \times 4 = 16 \).

So, after evaluating exponents for each part, the original mathematical expression \( \left(\frac{1}{2}\right)^{−2} \times\left(\frac{1}{3}\right)^{−2} \times\left(\frac{1}{4}\right)^{−2} \) simplifies to \( 4 \times 9 \times 16 \).

Performing the Final Calculation

Now we multiply the results from the previous step to find the final value of the expression. The calculation is \( 4 \times 9 \times 16 \).

First, multiply 4 by 9:

\( 4 \times 9 = 36 \)

Next, multiply this result by 16:

\( 36 \times 16 \)

We can perform this multiplication:

\( 36 \times 16 = 36 \times (10 + 6) = 36 \times 10 + 36 \times 6 \)

\( 36 \times 10 = 360 \)

\( 36 \times 6 = 216 \)

Adding these together:

\( 360 + 216 = 576 \)

Thus, the value of the expression \( \left(\frac{1}{2}\right)^{−2} \times\left(\frac{1}{3}\right)^{−2} \times\left(\frac{1}{4}\right)^{−2} \) after evaluating exponents and multiplying is 576.

This entire process demonstrates effectively evaluating exponents and simplifying mathematical expressions.