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Chapter 5: Problem 9

Use the product rule to simplify each expression. See Examples I and 2. $$ x^{5} \cdot x^{3} $$

Short Answer

Expert verified
The simplified expression is \(x^{8}\).

Step by step solution

01

Identify exponents of the base

In the given expression, the base is 'x', and the exponents are 5 and 3. The product rule for exponents tells us how to simplify expressions like this.
02

Apply the Product Rule

According to the product rule, when you multiply like bases, you add the exponents. For this expression it is \(x^{5} \cdot x^{3} = x^{5+3} = x^{8}\).
03

Write the simplified expression

After applying the product rule, the expression \(x^{5} \cdot x^{3}\) simplifies to \(x^{8}\).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions means reducing them to their most basic form without changing the value. It's like cleaning up a messy room; everything stays the same but looks much neater. In algebra, we simplify to make calculations easier and clearer.

There are some common steps you can follow:
  • Identify like terms or like bases. In our example, the like bases are the 'x's.
  • Use rules such as the product rule, which says to add exponents when multiplying like bases.
  • Check that everything is simplified to the fullest extent and no terms are left uncombined.
By following these steps, expressions can be made simpler, allowing for easier manipulation and understanding.
Exponents
Exponents are a way of expressing repeated multiplication of numbers or variables. They are the small numbers written to the upper right of a base number or variable. For example, in \(x^5\), '5' is the exponent and 'x' is the base. The exponent tells you how many times the base is used in multiplication.

Some basics about exponents to remember:
  • Any number (except zero) raised to the power of zero equals one, e.g., \(x^0 = 1\).
  • Increasing the exponent increases the value quickly, especially for larger bases.
  • Exponents of one simply mean the base itself, e.g., \(x^1 = x\).
Understanding these properties helps in applying rules like the product rule efficiently.
Algebra Concepts
Algebra is the branch of mathematics that involves variables and constants to represent numbers and expressions. It's like a game of finding the value of hidden numbers using rules and operations.

Here are a few key algebra concepts to keep in mind:
  • A variable represents unknown numbers and is usually a letter like 'x' or 'y'.
  • An expression is a combination of variables, numbers, and operations (like addition and multiplication).
  • Equations are statements that show two expressions are equal, generally connected by an '=' sign.
In algebra, learning to manipulate expressions using rules and properties is essential for solving equations and understanding mathematical relationships.

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Most popular questions from this chapter

Factor each polynomial completely. See Examples 1 through 12. $$ x^{2}+5 x+8 $$

Factor each polynomial completely. See Examples 1 through 12. $$ x^{2}+4 x-5 $$

Factor. Assume that variables used as exponents represent positive integers. $$ 3 x^{2 n}-8 x^{n}+4 $$

Without calculating, determine which number is larger. $$ 5^{-10} \text { or } 5^{-9} $$

Factor each polynomial completely. See Examples 1 through 12. $$ 16 y^{2}+6 y x-27 x^{2} $$
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