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When dealing with negative exponents, the first step is to understand the concept of the reciprocal. The reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 6 is \(\frac{1}{6}\), and the reciprocal of \(\frac{6}{7}\) is \(\frac{7}{6}\).
The reason we consider the reciprocal while solving negative exponents is because a negative exponent essentially tells us to 'flip' the base.
When you see \((\frac{6}{7})^{-2}\), you take the reciprocal of \(\frac{6}{7}\) to get \(\frac{7}{6}\). This step sets you up to work with a positive exponent in the next step, making the math much simpler.

After taking the reciprocal, the negative exponent changes to a positive exponent.
In our example, \(\frac{6}{7}^{-2}\) becomes \(\frac{7}{6}^2\). Now, you are ready to deal with a positive exponent, which is a more straightforward operation.
A positive exponent means that you'll be multiplying the base by itself as many times as the exponent specifies.
For instance, in \(\frac{7}{6}^2\), the 2 indicates that you should multiply \(\frac{7}{6}\) by itself:
\(\frac{7}{6} \times \frac{7}{6} = \frac{7^2}{6^2} = \frac{49}{36}\).
This conversion from a negative to a positive exponent simplifies our computations significantly.

The last important concept is simplifying fractions. After raising the reciprocal base to the positive exponent, you may get a fraction that can sometimes be simplified. In our case, \(\frac{7}{6}^2\) becomes \(\frac{49}{36}\).
Simplifying fractions is about reducing them to their simplest form. This involves finding the common factors of the numerator and the denominator and dividing them out.
For \(\frac{49}{36}\), the numbers 49 and 36 have no common factors other than 1, which means \(\frac{49}{36}\) is already in its simplest form.
This step ensures that your final answer is as straightforward and clean as possible.

• Always look for common factors.
• If there are none, your fraction is simplified.

Remember, clear and simplified fractions make your results easier to interpret and understand.