Problem 119 - Elementary Algebra

Question

Simplify the following problems. $$ x^{n} \cdot x^{m} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(x^{n} \cdot x^{m}\). Answer: The simplified expression is \(x^{n+m}\).

Step 1: Identify the Product of Powers rule

Since we have two terms with the same base (x) raised to different exponents (n and m), we can use the Product of Powers rule to simplify the expression. The rule states that \(a^{m} \cdot a^{n} = a^{m+n}\), where a is the base and m and n are the exponents.

Step 2: Apply the Product of Powers rule

Applying the Product of Powers rule to our expression, we get: $$ x^{n} \cdot x^{m} = x^{n+m} $$

Step 3: Write the final simplified expression

The simplified expression for the given problem is: $$ x^{n+m} $$

Key Concepts

Product of Powers Rule

When working with exponents, especially those involving variables, understanding the Product of Powers rule is essential. This rule states: when you multiply two powers with the same base, you simply add their exponents. In the formula, if you have a base \( a \) raised to an exponent \( m \), and the same base \( a \) raised to another exponent \( n \), multiplying these gives \( a^{m} \cdot a^{n} = a^{m+n} \).
This rule is incredibly useful when simplifying expressions, as it reduces potentially complicated multiplications to simpler calculations involving addition of the exponents.

• Base remains unchanged: Regardless of the exponents involved, the base stays the same throughout the process.
• Exponents are added: You simply add the exponent values together.

For instance, in the expression \( x^{3} \cdot x^{4} \), the base \( x \) remains the same, and the exponents 3 and 4 add up to give \( x^{7} \). This simplification allows for easier manipulation of algebraic expressions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They do not have an equals sign, differentiating them from algebraic equations. For example, \( 3x + 2y \) is an algebraic expression.
These expressions serve as building blocks for more complex mathematical operations in algebra. Variables, like \( x \) and \( y \), represent unknown values and allow for flexibility in problem-solving.

• Variables: Symbols used to denote unspecified numbers.
• Coefficients: Numbers that multiply the variables.
• Constants: Fixed numbers that stand alone without variables.

Understanding how to work with algebraic expressions is crucial for tasks such as solving equations, simplifying expressions, and even in calculus. Recognizing the structure helps in applying rules like the Product of Powers efficiently.

Simplification in Algebra

Simplifying algebraic expressions means making them as simple as possible while keeping their value unchanged. This process often involves using rules, such as the Product of Powers rule, to reduce complexity.
Simplification helps in reducing computational effort and can make solving equations much easier. A simplified expression is often more understandable and easier to work with.

• Combine like terms: This involves adding or subtracting terms that have the same variables raised to the same power.
• Apply exponent rules: Use rules like the Product of Powers to simplify products of the same variables.
• Reduce fractions: If the expression involves fractions, simplify them by dividing both the numerator and denominator by their greatest common factor.

For instance, the expression \( x^{2} + 4x + 4 - x^{2} \) can be simplified by combining like terms to get \( 4x + 4 \). By continually looking for opportunities to simplify, algebra becomes more manageable and less intimidating.