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Chapter 10: Problem 64

Simplify. Write the answers with positive exponents only. $$\left(p^{4}\right)^{-5}$$

Short Answer

Expert verified
The simplified answer is \(\frac{1}{p^{20}}\).

Step by step solution

01

- Understand the Problem

The expression given is \(\big(p^{4}\big)^{-5}\). The goal is to simplify this expression and write the answer with positive exponents only.
02

- Apply the Power Rule

Apply the power rule of exponents, which states that \( (a^m)^n = a^{m \times n} \). Here, \(a\) is \(p\), \(m\) is \(4\), and \(n\) is \(-5\). Therefore, \(\big(p^{4}\big)^{-5} = p^{4 \times (-5)} = p^{-20}\).
03

- Rewrite Using Positive Exponent

Rewrite \(p^{-20}\) using positive exponents. Recall that \(a^{-m} = \frac{1}{a^m}\). So, \(p^{-20} = \frac{1}{p^{20}}\). The final simplified form is \(\frac{1}{p^{20}}\).

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Key Concepts

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

Power Rule of Exponents
To simplify an expression involving exponents, the power rule of exponents is highly useful. This rule states \( (a^m)^n = a^{m \times n} \).
Let's break it down:
  • \
Negative Exponents
Negative exponents can be tricky, but they become much easier once you understand the key rule: \(a^{-m} = \frac{1}{a^m}\).
This means that if you have an exponent that's negative, you can rewrite the term as a fraction where the exponent becomes positive.
Consider the expression \p^{-20}\ from our exercise:
  • \
Positive Exponents
When simplifying expressions, it's often required to write your final answer with positive exponents.
This is especially true in most mathematical contexts and homework problems.
To convert a negative exponent to a positive exponent, recall the rule we've just practiced: \(a^{-m} = \frac{1}{a^m}\).
  • \

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Most popular questions from this chapter

Multiply the polynomials. $$\left(5 y^{2}-3 y-4\right)\left(y^{2}+4 y+7\right)$$

Simplify. Write the answer with positive exponents only. $$\frac{m^{-1} n^{7}}{p^{-3}}$$

$$\text { Fill in the blanks }\left(x^{3}\right)^{5}=x \square \cdot \square=x^{15}$$

Subtract the polynomials. $$\left(12 c^{2}+6 c-3\right)-\left(4 c^{2}-3 c+8\right)$$

Fill in the missing polynomial. $$(\quad)(z-5)=z^{2}-3 z-10$$
See all solutions

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