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Chapter 10: Problem 79

In the following exercises, simplify each expression using the Product Property of Exponents. $$x^{a} \cdot x^{2}$$

Short Answer

Expert verified
The simplified expression is \(x^{a+2}\).

Step by step solution

01

- Understand the Product Property of Exponents

The Product Property of Exponents states that when multiplying two expressions with the same base, you add the exponents. Mathematically, this is written as: \[x^m \times x^n = x^{m+n}\]
02

- Identify the Base and Exponents

In the given expression \(x^a \times x^2\), the base is \(x\). The exponents are \(a\) and \(2\).
03

- Apply the Product Property

Using the Product Property of Exponents, add the exponents together: \[x^a \times x^2 = x^{a+2}\]
04

- Simplify the Expression

The simplified expression is \[x^{a+2}\]

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Key Concepts

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

simplifying expressions
Simplifying expressions is a fundamental skill in algebra. To simplify an expression means to make it as straightforward as possible without changing its value. This often involves combining like terms, reducing fractions, or applying mathematical properties like exponent rules. Simplification helps in making calculations easier and expressions more readable. For example, consider the expression \(3x + 2x \). Since both terms contain the same variable, they can be combined: \[3x + 2x = 5x\]. By simplifying, we get a more concise expression.
exponent rules
Exponent rules are essential for working with algebraic expressions involving powers. They help simplify expressions with exponents. Key rules include:

  • Product Property: When multiplying two expressions with the same base, add the exponents: \[x^m \times x^n = x^{m+n}\].
  • Quotient Property: When dividing expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator: \[x^m \div x^n = x^{m-n}\].
  • Power Property: When raising a power to another power, multiply the exponents: \[(x^m)^n = x^{mn}\].
  • Zero Exponent: Any nonzero base raised to the power of zero is 1: \[x^0 = 1\].
  • Negative Exponent: A negative exponent represents the reciprocal of the base raised to the absolute value of the exponent: \[x^{-n} = \frac{1}{x^n}\].

By using these rules, you can simplify and solve complex algebraic expressions involving exponents.
algebra basics
Algebra forms the foundation of higher-level mathematics and is crucial for solving various real-world problems. Some basic concepts in algebra include:

  • Variables: Symbols used to represent unknown values. Common variables include x and y.
  • Constants: Fixed values that do not change.
  • Expressions: Combinations of variables, constants, and operators (such as +, -, *, /).
  • Equations: Statements that show the equality of two expressions, often including an unknown value to be solved for.
  • Operations: Basic arithmetic operations applied to algebraic expressions.

Understanding these fundamentals is vital as they are the building blocks for more complex algebraic operations and problem-solving. By mastering these basics, you will be better prepared for advanced topics in mathematics and science.

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Most popular questions from this chapter

A hiker drops a pebble from a bridge 240 feet above a canyon. The polynomial \(-16 t^{2}+240\) gives the height of the pebble \(t\) seconds a after it was dropped. Find the height when \(t=3\) .

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

Factor the greatest common factor from each polynomial. $$16 u-24$$

Simplify. $$y^{0}$$

Simplify. $$\frac{a^{6}}{a}$$
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