Problem 38 - Algebra A Combined Function

Question

Write with positive exponents. Simplify if possible. $$ \frac{1}{n^{-8 / 9}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( n^{8/9} \).

Step 1: Recognize the Negative Exponent

The expression given is \( \frac{1}{n^{-8/9}} \). \( n^{-8/9} \) has a negative exponent. Recall that a negative exponent indicates a reciprocal: \( a^{-b} = \frac{1}{a^b} \).

Step 2: Convert to Positive Exponent

Apply the rule for negative exponents. \( n^{-8/9} = \frac{1}{n^{8/9}} \). Substituting in the expression gives \( \frac{1}{n^{-8/9}} = \frac{1}{\frac{1}{n^{8/9}}} \).

Step 3: Simplify the Expression

Simplify \( \frac{1}{\frac{1}{n^{8/9}}} \) by recognizing it as a complex fraction. Simplify using the rule \( \frac{a}{\frac{b}{c}} = a \cdot \frac{c}{b} \). In this case, it simplifies to \( n^{8/9} \).

Key Concepts

Negative Exponents

When you see a negative exponent, it might look a bit confusing at first. The main thing to remember is that a negative exponent indicates something very special: an inverse or reciprocal.
For example, when you see something like \( n^{-b} \), it simply means that the base \( n \) is on the bottom of a fraction, and its exponent becomes positive: \( n^{-b} = \frac{1}{n^b} \).
This transformation allows us to convert any negative exponent into a more typical positive exponent form, making calculations much simpler.

• Understanding that negative exponents are not scary at all is key; it's just about flipping the base across the fraction line.
• By converting to positive exponents, you make the math easier to handle.

For example, in the expression \( n^{-8/9} \), instead of thinking of it as a complex operation, just remember it means \( \frac{1}{n^{8/9}} \). This will consistently help you anytime you encounter negative exponents.

Reciprocal Rule

The Reciprocal Rule is closely tied to negative exponents. It helps us understand how to simplify expressions, especially when dealing with fractions.
When a number or variable is raised to a negative exponent, it implies that the base is dividing 1.
So, if you see \( a^{-b} \), the reciprocal rule tells us that you can express this as \( \frac{1}{a^b} \). This swaps the position of the base \( a \) from the numerator to the denominator (or vice versa if needed).

• It's all about inverting the position of the base to deal with negative exponents effectively.
• For instance, \( n^{-8/9} \) becomes \( \frac{1}{n^{8/9}} \) using this rule.

This form simplifies further operations, especially when combined with other algebraic rules. Understanding the reciprocal rule makes negative exponent problems straightforward.

Simplifying Expressions

After dealing with negative exponents and applying the reciprocal rule, the next step is simplifying expressions. This often involves organizing and reducing mathematical expressions to their simplest form.
In our example, after changing \( n^{-8/9} \) to a positive exponent form \( \frac{1}{n^{8/9}} \), we find ourselves with a complex fraction \( \frac{1}{\frac{1}{n^{8/9}}} \). Simplifying expressions like these can be tricky, but there's a straightforward method: multiply by the reciprocal.

• Remember that for \( \frac{1}{\frac{b}{c}} \), you can simplify it by calculating \( c \cdot \frac{1}{b} \).
• By using this approach, \( \frac{1}{\frac{1}{n^{8/9}}} \) simplifies just to \( n^{8/9} \).

By reducing fractions and expressions to their simplest forms, we make it much easier to understand and use in further calculations. Simplifying expressions helps clear up any mathematical clutter and confusion.