91Ó°ÊÓ

Understanding reciprocals is essential when dealing with negative exponents. A reciprocal is simply the inverse of a number or fraction.
For a fraction, you swap the numerator (top part) with the denominator (bottom part). For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
We often find reciprocals when the exponent is negative.
For instance, in the exercise \( \frac{2}{3}^{-5} \), we take the reciprocal of \( \frac{2}{3} \), converting it to \( \frac{3}{2} \). This changes the negative exponent to a positive one, making the expression much simpler to handle.
By turning the negative exponent into a positive one, it becomes easier to perform further calculations.

Fraction exponentiation may seem daunting but is quite straightforward once you grasp the fundamentals. When you raise a fraction to a power, you apply the exponentiation to both the numerator and the denominator independently.
For instance, in our problem, \( \frac{3}{2}^5 \), we raise both 3 and 2 to the power of 5.
This gives us \( \frac{3^5}{2^5} \).
Breaking it down step by step:

• Calculate \( 3^5 = 243 \)
• Calculate \( 2^5 = 32 \)
• Combine them back to get \( \frac{243}{32} \)

Remember that exponents work the same way for fractions as they do for whole numbers. So once you raise both parts of the fraction, you simply combine them to get the final result.

Simplifying expressions is a crucial part of solving mathematical problems efficiently.
In our exercise, we start with a negative exponent and then simplify by converting to a reciprocal and applying the power rule.
Simplification often involves the following steps:

• Identify and apply exponent rules, such as changing a negative exponent to a positive one by taking the reciprocal.
• Perform the necessary calculations for exponentiation.
• Combine terms to finalize your result.

In our solution, we simplified \( \frac{2}{3}^{-5} \) by:

• Taking the reciprocal to get \( \frac{3}{2} \).
• Raising it to the power of 5 to get \( \frac{3^5}{2^5} \)
• Simplifying to \( \frac{243}{32} \)

By breaking down the problem into smaller parts and following these steps, you make complex problems more manageable and easier to solve.