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Chapter 6: Problem 99

Simplify each expression using the Product Property for Exponents. $$x^{4} \cdot x^{2}$$

Short Answer

Expert verified
x^{6}

Step by step solution

01

Identify the Product Property for Exponents

The Product Property for Exponents states that when you multiply two expressions with the same base, you add their exponents. Mathematically, this is expressed as: \[ a^m \times a^n = a^{m+n} \] where \(a\) is the base, and \(m\) and \(n\) are the exponents.
02

Apply the Product Property

Using the Product Property for Exponents, identify the base and the exponents in the given expression \[ x^4 \times x^2 \]. Here, the base \(x\) is the same, and the exponents are \(4\) and \(2\), respectively. Thus, add the exponents: \[ 4 + 2 \]
03

Simplify the Expression

Now, add the exponents: \[ x^{4+2} = x^{6} \]

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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 the same number or variable. For example, \( x^3 \) means \( x \) multiplied by itself three times (i.e., \( x \cdot x \cdot x \)). The number of times the base (in this case, \( x \)) is multiplied is called the exponent.
Here are some key points about exponents:
  • An exponent tells you how many times to use the base in a multiplication.
  • The base can be any real number or variable.
  • Exponents can be positive, negative, or zero.
Understanding exponents helps simplify complex algebraic expressions and solve problems more efficiently.
Simplifying Expressions
Simplifying expressions means making them as simple as possible. This often involves combining like terms and using mathematical properties, such as the Product Property for Exponents.
Here鈥檚 a breakdown:
  • Identify the base and exponents: Check if the bases are the same and then focus on the exponents.
  • Apply the relevant property: For our case, we use the Product Property for Exponents, which states that \( a^m \cdot a^n = a^{m+n} \).
  • Simplify: After applying the property, you add the exponents to simplify the expression fully.
For instance, in the given exercise, \( x^4 \cdot x^2 \), the base \( x \) is common, so we add the exponents (4 and 2) together to get \( x^6 \).
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In algebra, letters represent numbers, and these letters are called variables.
Key concepts in algebra include:
  • Expressions: Combinations of variables, numbers, and operations (like addition, subtraction, multiplication, and division). For example, \( 3x + 5 \) is an algebraic expression.
  • Equations: Statements that assert the equality of two expressions, usually involving an equals sign (\( = \)). For instance, \( x + 2 = 5 \).
  • Properties: Rules that describe how numbers and variables interact, such as the Product Property for Exponents.
Understanding algebra is essential, as it provides the foundation for more advanced mathematical concepts and real-world problem solving. Using properties like the Product Property for Exponents can simplify expressions and make solving equations much easier.

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