/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 98 Simplify each expression, if pos... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 98

Simplify each expression, if possible. $$ a b^{3} \cdot a b^{4} $$

Short Answer

Expert verified
The expression simplifies to \( a^2 b^7 \).

Step by step solution

01

Identify Like Bases

Observe that the expression involves two terms: \( a b^3 \) and \( a b^4 \). Both terms have the same base components \( a \) and \( b \).
02

Apply the Product Rule for Exponents

The product rule for exponents states that when multiplying like bases, you add the exponents. For the term \( a \), the exponent is implicitly 1 in each factor. So, \( a imes a = a^{1+1} = a^2 \). For the term \( b \), we have \( b^3 imes b^4 = b^{3+4} = b^7 \).
03

Combine the Simplified Terms

Combine the simplified parts, putting the results from Step 2 together: The expression becomes \( a^2 b^7 \).

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.

Product Rule for Exponents
When it comes to simplifying algebraic expressions, one useful tool is the product rule for exponents. This rule makes handling expressions with exponents much cleaner and more straightforward.

The product rule states that if you multiply two exponents with the same base, you simply add the exponents together. This is expressed mathematically as:
  • For two terms with like bases, such as \( x^m \) and \( x^n \), the result is \( x^{m+n} \).
When you see an expression like \( ab^3 \cdot ab^4 \), you're dealing with two like bases: both terms have a base of \( a \) and a base of \( b \).

By applying the product rule, you take the exponents of like bases and add them together to simplify the expression.
Like Bases
Identifying like bases is crucial when you're simplifying expressions with exponents. A 'like base' means the terms in the expression have the same base components.

In the expression \( ab^3 \cdot ab^4 \), both components \( a \) and \( b \) are bases that appear in each term.
  • Terms like \( a \times a \) are considered like bases and can be combined by adding their exponents.
  • Similarly, \( b^3 \times b^4 \) are like bases, allowing you to apply the product rule.
Recognizing which parts of your equation have like bases allows you to correctly apply rules and simplify the expression further. This recognition lays the groundwork to eventually combine and simplify the expression into its most concise form.
Exponent Addition
Once you've recognized like bases and applied the product rule, the next step is handling the addition of exponents. Exponent addition occurs because of the basic principle highlighted in the product rule.

Let's explore this within our earlier expression. For the base \( a \), each instance of \( a \) in the expression has an exponent of 1. So when you multiply \( a \times a \), it becomes \( a^{1+1} = a^2 \).

For the base \( b \), we have \( b^3 \times b^4 \). By adding the exponents, it becomes \( b^{3+4} = b^7 \). This efficient use of exponent addition through the product rule helps turn a complex expression into a much simpler one, like \( a^2b^7 \).
  • Remember: Always add exponents when multiplying like bases.
  • Exponent rules are powerful tools for expression simplification.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Perform the operations. $$ (3 m-2 n)^{3} $$

Perform the operations. $$ (3 x-2)^{2}+(2 x+1)^{2} $$

Perform each division. $$ \frac{10 n-6}{n} $$

Perform each division $$ \frac{-17 x^{2}+5 x+x^{4}+2}{4 x+x^{2}-1} $$

Perform each division. $$ \frac{12 x^{3} y^{2}-8 x^{2} y-4 x}{4 x y} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.