Problem 28 - Elementary and Intermediate Algebra

Question

Use the product rule for exponents to simplify each expression. Write the results using exponents. $$ 3^{4} \cdot 3^{6} $$

Step-by-Step Solution

Answer

The simplified expression is \(3^{10}\).

Step 1: Identify the Bases and Exponents

In the expression \(3^{4} \cdot 3^{6}\), identify the base and the exponents. The base for both terms is 3. The exponents are 4 and 6.

Step 2: Apply the Product Rule for Exponents

The product rule for exponents states that when multiplying two expressions with the same base, you can add the exponents. This can be written as \(a^m \cdot a^n = a^{m+n}\). Here, apply this rule to \(3^{4} \cdot 3^{6}\): \[3^{4+6} = 3^{10}\].

Step 3: Simplify the Expression

Now that the expression is \(3^{10}\), rewrite the original expression in its simplified form using this result. The simplified form is already in terms of exponents.

Key Concepts

Exponents

Exponents are powerful tools in mathematics that tell us how many times we need to multiply a number, known as the base, by itself. Consider the exponent in the expression \(3^4\). Here, the base is 3, and the exponent is 4, which means 3 is multiplied four times: \(3 \cdot 3 \cdot 3 \cdot 3\). This makes calculations easier when dealing with large numbers or repeated multiplication. Exponents follow specific rules that help simplify complex expressions. One of these is the product rule, which we delve into when simplifying expressions.

Simplifying Expressions

Simplifying expressions means making them easier to work with or understand by reducing them to their most concise form. When simplifying expressions with exponents, we often use exponent rules such as the product rule. This particular rule is used when we multiply terms with the same base. For example, with \(3^4 \cdot 3^6\), the product rule states that you can add the exponents since the base is common. This simplifies the expression to \(3^{4+6} = 3^{10}\). By applying these rules, we can transform complex expressions into simpler ones that are easier to handle in equations and other operations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and exponents structured within mathematical operations like addition, subtraction, multiplication, and division. These expressions may appear complex, but learning which rules to apply makes them manageable. In the expression \(3^4 \cdot 3^6\), we focus primarily on the exponents and their management through rules like the product rule. In algebra, it is essential to simplify expressions to solve equations or find values efficiently. When you understand how to manipulate these expressions using properties like those of exponents, you open up a greater ability to solve algebraic problems effectively.