Problem 19 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. \(4^{3} \cdot 4^{6}\)

Step-by-Step Solution

Answer

Simplified expression is \(4^{9}\).

Step 1: Recognize the base

Recognize the expressions \(4^{3}\) and \(4^{6}\) have the same base, which is 4.

Step 2: Apply the property of exponents

For multiplication with the same base, the exponents get added. So, apply this property here and add the exponents of the two terms.

Step 3: Calculate the sum of exponents

The sum of \(3\) and \(6\) is \(9\). So, \(4^{3}\cdot4^{6}=4^{9}\).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial for manipulating and simplifying expressions that involve exponents.

When multiplying exponential expressions with the same base, the property to apply is the Product of Powers. This property states that when you multiply two exponents with the same base, you can simply add the exponents. Mathematically, this is represented as \( a^m \cdot a^n = a^{m+n} \).

This property also helps when dividing exponents with the same base (Quotient of Powers), or raising a power to another power (Power of a Power). These properties ease the process of working with complex algebraic and exponential expressions.

Simplifying Exponential Expressions

To simplify exponential expressions, one must follow certain algebraic rules and properties of exponents. Simplification becomes easier when you recognize patterns like common bases. As seen in the example \(4^3 \cdot 4^6\), the base is the same, so we apply the Product of Powers rule.

Other essential steps include combining like terms, using the Power of a Product rule where each factor inside the parentheses is raised to the exponent, and applying the Zero-Exponent rule, where any non-zero base with an exponent of zero equals one. Simplification is a systematic process, and each step's accuracy ensures the integrity of the solution.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. The process of manipulating these expressions according to algebraic rules forms a fundamental part of algebra.

In the context of exponential expressions, it's important to recognize the parts: the base, which is the number getting multiplied; and the exponent, which tells us how many times to multiply the base by itself. The expression \(4^{3} \cdot 4^{6}\) demonstrates how one might encounter algebraic expressions involving exponents in their studies.

Exponential Functions

Exponential functions are a type of algebraic function represented by the formula \(f(x) = a \cdot b^{x}\), where \(a\) is a constant, \(b\) is the base, and \(x\) is the exponent. The base \(b\) is a positive real number not equal to 1, and the exponent \(x\) is a variable.

These functions are known for their rapid growth or decay and are commonly used in contexts like population growth, radioactive decay, and compounding interest. Exponential functions can be graphed on a coordinate plane, showing a curve that increases or decreases depending on the value of the base \(b\).