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Chapter 10: Problem 369

In the following exercises, simplify. $$\left(p^{-3}\right)^{2}$$

Short Answer

Expert verified
\( \left(p^{-3}\right)^2 = \frac{1}{p^6} \)

Step by step solution

01

Understand the Expression

The given expression is \(\big(p^{-3}\big)^2\). It involves an exponent being raised to another exponent.
02

Apply the Power Rule

Recall the power rule which states that \(\big(a^m\big)^n = a^{m \times n}\). Applying this to our expression, we get \(\big(p^{-3}\big)^2 = p^{-3 \times 2}\).
03

Simplify the Exponent

Multiply the exponents: \(-3 \times 2 = -6\). Thus, the expression simplifies to \(p^{-6}\).
04

Interpret the Negative Exponent

A negative exponent indicates the reciprocal. Therefore, \(p^{-6} = \frac{1}{p^6}\).

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Key Concepts

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

Understanding Negative Exponents
Negative exponents can initially seem confusing, but they are straightforward once you grasp the basic idea. When you see an exponent with a negative sign, it means that you are dealing with the reciprocal of the base raised to the opposite positive exponent. For example, consider the expression \(p^{-6}\). Here, the base is \(p\) and the exponent is \(-6\). We can rewrite this expression as \(\frac{1}{p^6}\). In essence, a negative exponent does not imply a negative number. Instead, it tells us to take the 'one over' the number.

To summarize:
  • Negative exponent \(a^{-b}\) becomes the reciprocal: \(\frac{1}{a^b}\).
  • This allows us to work with positive exponents, making calculations easier.
Applying the Power Rule
The power rule is a key property of exponents that simplifies complex expressions involving exponents. The power rule states: \( (a^m)^n = a^{m \times n} \). This means that when you raise a power to another power, you multiply the exponents.

Let's apply this rule to our example expression: \(\big(p^{-3}\big)^2\). According to the power rule, you multiply the exponents: \( -3 \times 2 \), which gives \( p^{-6} \).
In essence, the power rule helps us combine exponents efficiently and is critical for simplifying expressions with nested exponents.
  • When applying the power rule, always multiply the exponents.
  • Watch out for negative exponents and handle their simplification separately.
Understanding Reciprocals
Reciprocals are essentially the 'flip' of a number. For any non-zero number \(a\), its reciprocal is \(\frac{1}{a}\). Reciprocals play a significant role in simplifying negative exponents. For instance, we've seen that \(p^{-6} = \frac{1}{p^6}\). This flips the base and turns the exponent positive.

Reciprocals are useful in various aspects of algebra and help manage complex expressions smoothly. You can always revert a reciprocal operation by taking the reciprocal again.
  • Reciprocal of \(a\) is \( \frac{1}{a} \).
  • Negative exponents invoke the reciprocal function.

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