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Fraction simplification is an important skill in mathematics, allowing you to reduce fractions to their simplest form. This process makes them easier to understand and work with. In the exercise, the final simplification occurs after flipping the fraction due to the negative exponent rule. You start with the expression \(\frac{1}{\frac{2}{3}}\). To simplify this, you need to multiply both the numerator and the denominator by the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\). This results in:

• Multiply the numerator: \(1 \times \frac{3}{2} = \frac{3}{2}\)
• Multiply the denominator: \(\frac{2}{3} \times \frac{3}{2} = 1\)

As the denominator becomes 1, the fraction simplifies to \(\frac{3}{2}\). This simplification reveals that the negative exponent effectively changes the fraction to its reciprocal.

Exponent rules are fundamental in dealing with powers and exponents and are particularly helpful when simplifying expressions that involve powers. One crucial rule is the "negative exponent rule". This rule tells us that a negative exponent essentially "flips" the base into its reciprocal. For instance, the expression

• \(a^{-n} = \frac{1}{a^n}\),

means any non-zero base \(a\) raised to a negative exponent \(-n\) is equivalent to 1 divided by \(a\) raised to the positive exponent \(n\). In the given problem, you identify \(\left(\frac{2}{3}\right)^{-1}\) and apply this rule, effectively taking the reciprocal of \(\frac{2}{3}\). Understanding these rules helps you manage and simplify more complex algebraic expressions accurately.

Inverse operations are operations that reverse the effect of another operation. In the context of the original exercise, the concept of inverse operations plays a key role through the idea of reciprocals and negative exponents. When you encounter a fraction raised to a negative exponent, such as \(\left(\frac{2}{3}\right)^{-1}\), you use the inverse operation of taking the reciprocal. This means instead of dividing by \(\frac{2}{3}\), you multiply by its inverse, \(\frac{3}{2}\). This approach is visually represented as flipping the fraction upside down:

• Original fraction: \(\frac{2}{3}\)
• Inverse operation: \(\frac{3}{2}\)

By applying this inverse, you simplify the expression according to the exponent rules, translating the use of a negative exponent into a practical operation of reciprocal transformation.