Problem 61 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. $$ x^{3} \cdot x^{5} $$

Step-by-Step Solution

Answer

The expression simplifies to \(x^{8}\).

Step 1: Identify the bases and exponents

In the given expression \(x^{3} \cdot x^{5}\), the base is \(x\) and the exponents are 3 and 5.

Step 2: Apply the rule of multiplying powers with the same base

When multiplied, powers with the same base add up. In this case, add the exponents 3 and 5 together to get 8.

Step 3: Write the expression as a single power

After adding the exponents, the expression simplifies to \(x^{8}\). This is the expression written as a single power of the base.

Key Concepts

Power of a Product

One essential concept in algebra is understanding the "power of a product." When you have an expression like \(x^3 \cdot x^5\), both terms have the same base, which is \(x\). The power of a product rule states that when you multiply exponents with the same base, you simply add the exponents together. It's like combining strengths in a team, each exponent contributes to the overall product.

• The formula for the power of a product when bases are the same is: \(b^m \cdot b^n = b^{m+n}\).
• In our example, we apply this rule by adding 3 and 5, leading to \(x^{3+5}\) or \(x^8\).
• This shows the expression as a single power: the final answer is \(x^8\).

Now, whenever you encounter similar terms with the same base, remember this rule to quickly simplify the expressions by adding exponents.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They are like a puzzle, where we need to find how all pieces fit together. In the expression \(x^3 \cdot x^5\), we see two such terms connected with multiplication.

• The variable part in these expressions is \(x\), which serves as the common base.
• Each \(x\) is raised to a power, showing how many times the base is used in multiplication.
• Understanding the structure of algebraic expressions helps us in identifying how to apply the correct rules, such as the power of a product.

Algebraic expressions can look complicated at first, but breaking them down step by step—recognizing bases and powers—makes them manageable.

Simplifying Expressions

"Simplifying expressions" means transforming them into their simplest or most efficient form. This process makes work with algebraic equations easier and clearer.

• With the expression \(x^3 \cdot x^5\), simplifying involves applying known rules—like adding exponents when multiplying bases that are the same.
• The key objective is to express the result in a concise way, as \(x^8\), making the expression easier to handle in mathematical operations.
• This practice not only saves time but also reduces the chances of errors in more complex computations.

By mastering the simplification of expressions, students gain confidence in observing algebraic patterns and applying rules. Understanding these foundational processes is what makes simplifying expressions so valuable in mathematics.