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Chapter 10: Problem 359

In the following exercises, simplify. $$\left(p q^{-4}\right)\left(p^{-6} q^{-3}\right)$$

Short Answer

Expert verified
\(\frac{1}{p^{5} q^{7}}\)

Step by step solution

01

Distribute the Exponents

Apply the property of exponents that allows distribution over multiplication. Rewrite the expression as \(\begin{array}{c}\frac{p q^{-4}}{1}\times \frac{p^{-6} q^{-3}}{1} \end{array} \).
02

Combine the Bases with the Same Base

When multiplying like bases, add the exponents of each base: for \(p^{1}\times p^{-6}\rightarrow p^{1 + (-6)} = p^{-5}\) and \(q^{-4}\times q^{-3}\rightarrow q^{-4 + (-3)} = q^{-7}\). So, the expression becomes \(p^{-5}q^{-7}\).
03

Simplify the Expression

Since \(p^{-5}\rightarrow \frac{1}{p^{5}}\) and \(q^{-7}\rightarrow \frac{1}{q^{7}}\), the expression becomes \(\frac{1}{p^{5} q^{7}} \).

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Key Concepts

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

Exponents
Exponents are a way to express repeated multiplication of a number by itself. For example, in the term \(a^n\), \(a\) is the base and \(n\) is the exponent. The exponent tells us how many times to multiply the base by itself. For instance, \(2^3\) means \(2 \times 2 \times 2 = 8\).

There are key properties of exponents that make simplifying expressions easier:
  • Product of Powers Rule: When multiplying like bases, add their exponents. For instance, \(a^m \times a^n = a^{m+n}\).
  • Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, \((a^m)^n = a^{m \times n}\).
  • Power of a Product Rule: When raising a product to a power, apply the exponent to each factor. For example, \((ab)^n = a^n \times b^n\).
Understanding and applying these rules helps in simplifying complex algebraic expressions.
Multiplication of Polynomials
Multiplication of polynomials involves distributing each term of one polynomial to every term of the other polynomial. This process often requires applying the distributive property, which is essential in expanding expressions.

Here are the steps involved:
  • Distribute each term: Multiply each term in the first polynomial by each term in the second polynomial.
  • Combine like terms: After distribution, collect and combine terms that have the same variables and exponents.
For example, consider multiplying \((p q^{-4})\) and \((p^{-6} q^{-3})\). First, distribute to get each combination of the terms multiplied:

\((p \times p^{-6}) (q^{-4} \times q^{-3})\)

Apply the product of powers rule to combine like bases:
\(p^{1 + (-6)} q^{-4 + (-3)} = p^{-5} q^{-7}\).

The final step is simplifying if needed, which can involve converting negative exponents to fractions.
Negative Exponents
Negative exponents represent the reciprocal of the base raised to the opposite positive exponent. For example, \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\). This property is crucial for simplifying expressions with negative exponents.

Here鈥檚 how to handle negative exponents
  • Rewrite negative exponents as fractions: For any term with a negative exponent, take the reciprocal and change the exponent to positive. For instance, \(p^{-5}\) becomes \(\frac{1}{p^5}\) and \(q^{-7}\) becomes \(\frac{1}{q^7}\).
  • Combine fractions: Once rewritten, it's easier to combine and simplify expressions.
Using our exercise example, \(p^{-5} q^{-7}\) simplifies to:

\(\frac{1}{p^5} \times \frac{1}{q^7} = \frac{1}{p^5 q^7}\).

This final form is often the simplest way to represent terms with negative exponents.

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Most popular questions from this chapter

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