Problem 23 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. \(x^{6} \cdot x^{3}\)

Step-by-Step Solution

Answer

\(x^{9}\)

Step 1: Identify the Base and Exponents

The base in the given expression \(x^{6} \cdot x^{3}\) is \(x\), and the exponents are 6 and 3, respectively.

Step 2: Apply the Rule of Exponents for Multiplication

To simplify the expression, we use the rule of exponents for multiplication which is \(a^{m} \cdot a^{n} = a^{m+n}\). Here, \(a\) represents the base and \(m\) and \(n\) are the exponents.

Step 3: Perform the Calculation

Following step 2, we add the exponents together while keeping the base constant. Therefore, the expression \(x^{6} \cdot x^{3}\) simplifies to \(x^{6+3}\), or, \(x^{9}\).

Key Concepts

Exponential Expressions

Exponential expressions are mathematical notations used to represent numbers in terms of a base raised to a power, known as an exponent. An exponential expression is written as \(a^n\), where \(a\) is the base and \(n\) is the exponent. This format indicates that the base \(a\) is multiplied by itself \(n\) times.

• The base is the number or variable that is being multiplied.
• The exponent indicates how many times the multiplication occurs.

For example, in the expression \(x^6\), \(x\) is the base being multiplied by itself 6 times. Recognizing exponential expressions is crucial because they form the foundation of many math operations and help simplify complex calculations.

Product of Powers

Understanding the product of powers rule is essential when dealing with exponential expressions. This rule helps us multiply expressions that have the same base. The product of powers rule states: if you are multiplying two exponential expressions with the same base \(a\), you can add their exponents: \(a^m \cdot a^n = a^{m+n}\).

• Ensure the bases are the same before using this rule.
• Add only the exponents and keep the base unchanged.

For instance, in the expression \(x^6 \cdot x^3\), since both terms have the base \(x\), we simply add the exponents 6 and 3 to get \(x^{6+3}\). Applying this rule helps streamline computations by reducing the number of multiplication steps necessary.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest or most manageable form, keeping calculations clear and concise. When working with exponential expressions, simplifying often means applying rules, like the product of powers, to combine terms effectively. In our example \(x^6 \cdot x^3\), simplification uses the product of powers rule to add the exponents, resulting in \(x^9\).

• Always check expressions for common bases to apply exponent rules effectively.
• Simplified forms help in further calculations and provide clearer results.

By simplifying, we achieve a form that is more efficient for subsequent operations, making mathematical expressions easier to handle and understand.