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Chapter 10: Problem 352

In the following exercises, simplify. $$p^{5} \cdot p^{-2} \cdot p^{-4}$$

Short Answer

Expert verified
\(p^{-1}\) or \(\frac{1}{p}\)

Step by step solution

01

Identify the property of exponents

When multiplying terms with the same base, you add the exponents. The property is: \(a^{m} \times a^{n} = a^{m+n}\).
02

Apply the property to combine the exponents

Combine the exponents of \(p\) by adding them together:\(p^{5} \times p^{-2} \times p^{-4} = p^{5 + (-2) + (-4)}\).
03

Simplify the exponent

Add the exponents: \(5 + (-2) + (-4) = 5 - 2 - 4 = -1\).
04

Write the simplified expression

The simplified expression is:\(p^{-1}\).
05

Express the result with a positive exponent if needed

To express \(p^{-1}\) with a positive exponent, write it as: \(\frac{1}{p}\).

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Key Concepts

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

Properties of Exponents
Understanding the properties of exponents is essential when simplifying expressions involving powers. One key property to remember is that when you multiply terms with the same base, you add their exponents. This property can be written as:

\(a^m \times a^n = a^{m+n} \).

For example, if you have \(p^5 \times p^{-2} \times p^{-4}\), you add the exponents 5, -2, and -4. This gives you

\(p^{5-2-4} = p^{-1}\).

Recognizing and applying this property helps simplify expressions efficiently and accurately. It鈥檚 crucial to practice using this property so it becomes second nature.
Combining Like Terms
When simplifying algebraic expressions, combining like terms is a key step. Like terms are terms that have the same variable raised to the same power. For instance, in the expression \(p^5 \times p^{-2} \times p^{-4}\), all terms involve the variable \(p\).

To combine these terms, you use the properties of exponents. Specifically, you add the exponents of like terms.

Example:

\(p^5 \times p^{-2} \times p^{-4}\) becomes \(p^{5-2-4} = p^{-1}\).

Combining like terms in this way simplifies the expression, making it easier to work with.
Negative Exponents
Negative exponents can be confusing, but they follow specific rules. A negative exponent indicates that the base should be taken as the reciprocal. For example, \(p^{-1}\) can be written as \(\frac{1}{p}\).

Here鈥檚 a general rule: \(a^{-m} = \frac{1}{a^m}\). This means that any term with a negative exponent can be rewritten as a fraction.

In our simplified expression \(p^{-1},\) we apply this rule:

\(p^{-1} = \frac{1}{p}\).

Understanding this concept is important for expressing your answers in different forms. Practice with different problems will help you master the use of negative exponents.

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Most popular questions from this chapter

Multiply using any method. $$(5 b-2)\left(3 b^{2}+b-9\right)$$

Convert each number to decimal form. $$3.75 \times 10^{-1}$$

Multiply the binomials using various methods. $$(p+4)(p+7)$$

Simplify. $$\frac{a^{6}}{a}$$

Divide the monomials. $$72 p^{12} \div 8 p^{3}$$
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