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Chapter 5: Problem 60

Simplify each expression. Write answers using positive exponents. $$ n^{-9} \cdot n^{-2} $$

Short Answer

Expert verified
The simplified expression is \( \frac{1}{n^{11}} \).

Step by step solution

01

Apply the Product of Powers Rule

When multiplying expressions with the same base, add the exponents. Our expression is \( n^{-9} \cdot n^{-2} \). According to the rule, \( n^a \cdot n^b = n^{a+b} \). Therefore, \( n^{-9} \cdot n^{-2} = n^{-9 + (-2)} \).
02

Perform the Exponent Addition

Calculate the sum of the exponents from the previous step: \(-9 + (-2) = -11\). Therefore, we have \( n^{-11} \).
03

Convert Negative Exponents to Positive Exponents

To express \( n^{-11} \) with a positive exponent, use the property \( n^{-a} = \frac{1}{n^a} \). Applying this to our result gives \( n^{-11} = \frac{1}{n^{11}} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product of Powers Rule
When multiplying powers with the same base, the exponents are added together. This is known as the Product of Powers Rule. It provides a handy shortcut that simplifies expressing powers. For instance, with our exercise, you are dealing with terms like \( n^{-9} \cdot n^{-2} \). Both terms share the base \( n \).
To simplify, add the exponents:
  • Apply the rule: \( n^a \cdot n^b = n^{a+b} \).
  • So, \( n^{-9} \cdot n^{-2} = n^{-9+(-2)} \).
  • This results in \( n^{-11} \).
Adding exponents is all about combining the numbers through simple addition, even when they are negative.
Negative Exponents
Understanding negative exponents is crucial in mathematics. A negative exponent indicates the reciprocal of the base raised to the positive of that exponent.
To illustrate, let's take the result from our previous step: \( n^{-11} \). A negative exponent transforms in the following manner:
  • Use the rule: \( n^{-a} = \frac{1}{n^a} \).
  • So, \( n^{-11} \) becomes \( \frac{1}{n^{11}} \).
Negative exponents can make things look complex initially, but they are simply a way to express fractions or reciprocals. Recognize this pattern, and they become much easier to handle.
Simplifying Expressions
Simplifying expressions is all about making them easier to understand and work with. When simplifying, you usually aim to write the expression in a more manageable format.
For expressions using exponents, you:
  • Apply exponent rules like the Product of Powers Rule to simplify calculations
  • Convert negative exponents to positive by taking the reciprocal
  • Present the result in the simplest form with positive exponents
In our exercise, simplifying started with combining exponents and ended with changing negative to positive exponents, transforming \( n^{-11} \) into \( \frac{1}{n^{11}} \). This step-by-step progression helps ease the process of handling exponential expressions, making them more digestible.

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