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Chapter 10: Problem 57

Simplify. Write the answers with positive exponents only. $$p^{-8} p^{3}$$

Short Answer

Expert verified
\[ \frac{1}{p^5} \].

Step by step solution

01

- Understanding the Problem

The expression given is \( p^{-8} p^{3} \). To simplify this, use the rules of exponents.
02

- Apply the Product of Powers Rule

When multiplying terms with the same base, add the exponents. Hence, \(p^{-8} p^{3} = p^{-8+3} = p^{-5} \).
03

- Convert to Positive Exponent

To express this with a positive exponent, recall that \(a^{-n} = \frac{1}{a^n}\). Therefore, \( p^{-5} \) can be written as \frac{1}{p^5} \.

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Key Concepts

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

Negative Exponents
Negative exponents might seem confusing at first, but they are quite simple once you understand them.
In general, a negative exponent indicates that the base is on the wrong side of a fraction line.
For instance, if you have \(a^{-n}\), you can rewrite it as \(\frac{1}{a^n}\).
This means that instead of having \(a\) multiplied by itself \(n\) times, you take the reciprocal and have \(1\) divided by \(a\) multiplied by itself \(n\) times.
So, \(p^{-5}\) becomes \(\frac{1}{p^5}\). Essentially:
  • Negative exponent means 'one over' the positive exponent.

Understanding this will help you handle and simplify expressions with negative exponents easily.
Product of Powers Rule
The product of powers rule is a fundamental concept when dealing with exponents.
It tells you how to handle the multiplication of terms that have the same base but different exponents.
According to this rule, when you multiply two exponents with the same base, you simply add the exponents.
This can be written as \(a^m \times a^n = a^{m+n}\).
So for the expression \(p^{-8} p^{3}\), you add the exponents together:
  • \(-8 + 3 = -5\).
This simplifies the expression to \(p^{-5}\).
Knowing this rule is essential for simplifying more complex exponent expressions.
Positive Exponents
Finally, let's talk about positive exponents.
Positive exponents are straightforward since they indicate how many times you multiply a base by itself.
For example, \(a^3\) means \(a \times a \times a\).
When you need to convert a negative exponent into a positive one, remember to use the reciprocal.
So \(p^{-5}\) becomes \(\frac{1}{p^5}\).
By converting negative exponents to positive forms, you often make expressions easier to understand and work with.
For the given problem, we converted \(p^{-5}\) to \(\frac{1}{p^5}\) to express it with a positive exponent.
This step ensures your final answer aligns with standard mathematical conventions and is easier to interpret.

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Most popular questions from this chapter

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