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Chapter 2: Problem 25

Simplify using the quotient rule. $$\frac{a^{-1}}{a^{9}}$$

Short Answer

Expert verified
The simplified expression using the quotient rule is \(a^{-10}\).

Step by step solution

01

Identify the exponents

The given expression is \(\frac{a^{-1}}{a^9}\), where the exponent of the numerator is -1 and the exponent of the denominator is 9.
02

Apply the quotient rule

According to the quotient rule, when dividing two exponential expressions with the same base, we subtract the exponent of the denominator from the exponent of the numerator. In this case, we will subtract 9 from -1: \(-1 - 9 = -10\)
03

Simplify the expression

Now that we have the new exponent, we can rewrite the expression as follows: \(a^{-1-9}\) \(a^{-10}\) So, the simplified expression using the quotient rule is \(a^{-10}\).

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Key Concepts

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

Understanding Exponents
Exponents are an essential concept in algebra that help express repeated multiplication in a compact form. When you see something like \(a^n\), it means that \(a\) is multiplied by itself \(n\) times. The base \(a\) can be any number, while the exponent \(n\) indicates how many times the base is used as a factor.
  • Positive exponents (e.g., \(a^3\)) tell you to multiply the base by itself a certain number of times (\(a \times a \times a\)).
  • Negative exponents (e.g., \(a^{-2}\)) imply the reciprocal of the base raised to the corresponding positive exponent (\(\frac{1}{a^2}\)).
  • An exponent of zero (\(a^0\)) means the result is always 1, regardless of the base.
Understanding how to manipulate exponents is crucial when simplifying expressions and performing algebraic operations.
Simplifying Expressions with the Quotient Rule
Simplifying expressions is the process of reducing them to their simplest form. This often involves applying mathematical rules, such as the quotient rule for exponents, to achieve a concise expression.
The quotient rule is particularly useful when dividing exponential expressions with the same base. It suggests that we take the exponent of the numerator and subtract the exponent of the denominator:
  • Formula: \(\frac{a^m}{a^n} = a^{m-n}\)
  • Example: For \(\frac{a^{-1}}{a^9}\), identify the exponents: -1 for the numerator and 9 for the denominator. Then subtract: \(-1 - 9 = -10\).
  • Result: Simplifying gives \(a^{-10}\).
This results in the expression being transformed into a single term with a new exponent. It's important because it makes complex expressions more manageable and easier to work with.
Exploring Algebraic Expressions
Algebraic expressions form the backbone of algebra and consist of variables, constants, and operations. They are mathematical phrases that can represent a variety of computations. Each component of an algebraic expression has a role to play:
  • Variables: Often represented by letters like \(a, b, x,\) and \(y\), they stand in place of numbers that can change or do not have a fixed value.
  • Constants: These are numbers with a fixed value, such as 2 or -3, which do not change.
  • Operations: Include addition, subtraction, multiplication, and division, which determine how variables and constants are combined.
Understanding algebraic expressions involves a keen comprehension of how to manipulate these parts to solve equations, simplify expressions, and ultimately find solutions to mathematical problems. Mastery of these expressions paves the way for more advanced study in subjects requiring algebra.

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Most popular questions from this chapter

Simplify the expression using the product rule. Leave your answer in exponential form. $$7^{5} \cdot 7 \cdot 7^{4}$$

Perform the operation as indicated. Write the final answer without an exponent. \(\frac{2 \times 10^{1}}{5 \times 10^{4}}\)

Simplify the expression using the product rule. Leave your answer in exponential form. $$(-3) \cdot(-3)^{5} \cdot(-3)^{2}$$

Simplify using the quotient rule. $$\frac{6 v^{1} w}{54 v^{2} w^{-5}}$$

Simplify using the quotient rule. $$\frac{b^{5}}{b^{-1}}$$
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