Problem 595 Simplify. $$r^{-5} \cdot r^{-4... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 595

Simplify. $$r^{-5} \cdot r^{-4}$$

Short Answer

Expert verified

r^{-9}

Step by step solution

Identify the Exponents

Notice that the expression involves the base 'r' with exponents -5 and -4. The expression is $ r^{-5} \cdot r^{-4} $

Apply Exponent Addition Rule

When multiplying expressions with the same base, add the exponents. Combine the exponents: $ -5 + (-4) = -9 $

Rewrite the Expression

Substitute the sum of the exponents back into the expression: $ r^{-9} $

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Exponent Rules

Exponent rules are essential to simplify expressions that involve powers. Here are the basic rules:

**Product of Powers Rule**: When you multiply two expressions with the same base, you add the exponents: $a^m \times a^n = a^{m+n} $.
**Power of a Power Rule**: When you raise an exponent to another exponent, you multiply the exponents: $(a^m)^n = a^{mn} $.
**Quotient of Powers Rule**: When you divide two expressions with the same base, you subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.
**Power of a Product Rule**: When you raise a product to an exponent, distribute the exponent to each factor: $(ab)^n = a^n b^n $.

These rules simplify the process of working with exponents and are the foundation for all exponent-related operations.
In this exercise, we focus on the Product of Powers Rule because we are multiplying expressions with the same base.

Negative Exponents

Negative exponents might seem tricky, but they actually represent the reciprocal of the base raised to the positive exponent.
In simpler terms:

$a^{-n} = \frac{1}{a^n} $
$\frac{1}{a^{-n}} = a^n $.

So, a negative exponent doesn't mean the value is negative; instead, it means the value is a fraction.

For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.
In our exercise, the product $r^{-5} \times r^{-4}$ involves negative exponents.
We apply the rule for multiplying exponents, and the final answer, $r^{-9}$, means $\frac{1}{r^9}$. It flips to the reciprocal of $r$ raised to the positive power.

Multiplying Exponents

Multiplying exponents involves using the product rule we mentioned earlier.
Here鈥檚 a step-by-step of our exercise for clarity:

Expression: $r^{-5} \times r^{-4}$.
Identify the bases and exponents: Both terms have the same base 'r' with exponents -5 and -4.
Apply the product of powers rule: Add the exponents together: $-5 + (-4) = -9$.
Combine the result with the base: $r^{-9}$.

In practice:

$r^{2} \times r^{3} = r^{2+3} = r^{5}$.
$x^{6} \times x^{-2} = x^{6-2} = x^{4}$.

Understanding these steps will help simplify any exponent expressions more efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Simplify. $$r^{-5} \cdot r^{-4}$$

Short Answer

Step by step solution

Identify the Exponents

Apply Exponent Addition Rule

Rewrite the Expression

Key Concepts

Exponent Rules

Negative Exponents

Multiplying Exponents

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Discrete Mathematics

Calculus

Probability and Statistics

Geometry

Study anywhere. Anytime. Across all devices.