Problem 39 - College Algebra

Question

Simplify each exponential expression $$ \left(8 x^{3}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified form of the exponential expression \((8x^{3})^{2}\) is \(64x^{6}\).

Step 1: Identify the base and the exponent

In the given exponential expression, the base is \(8x^{3}\) which is inside parentheses and the exponent is 2, which is outside the parentheses.

Step 2: Apply the power of a power rule

According to the power of a power rule, we are able to raise each part of the base \(8x^{3}\) to the power of 2 separately. This means: \((8x^{3})^{2} = 8^{2} * (x^{3})^{2}\).

Step 3: Simplify the exponential terms separately

After applying the rule, the expression becomes \(64x^{6}\). Here, \(8^{2}\) simplifies to 64 and the simplification of \((x^{3})^{2}\) gives \(x^{6}\), which is derived from the power of a power rule where the powers are multiplied together. Hence, the final simplified form of the expression is \(64x^{6}\).

Key Concepts

Understanding the Power of a Power Rule

The power of a power rule is a fundamental concept in dealing with exponential expressions. This rule helps simplify expressions where one exponential term is inside another exponent, like \((a^m)^n\). According to this rule:

• When you have a base raised to an exponent, and that whole expression is raised to another exponent, you multiply the exponents together.
• Mathematically, \((a^m)^n = a^{m \times n}\).

This rule is immensely helpful when simplifying complex exponential expressions as it reduces the number of steps needed.
In our example of \((8x^3)^2\), each part inside the parenthesis is addressed separately according to the rule. That means: \(8\) gets squared and \(x^3\) is raised to the power of 2. This results in \(8^2\) and \((x^3)^2\), simplifying the process to manage and solve exponential expressions.

Simplifying Exponents Step by Step

Simplifying exponents refers to the process of reducing an expression with exponents to its simplest form. This often involves several key ideas and steps:

• Use rules like the power of a power rule to streamline your expressions.
• Handle coefficients (numbers) and variables separately, applying rules relevant to each.

In the expression \((8x^3)^2\), simplification involves:1. Squaring the coefficient: \(8^2 = 64\).2. Applying the power of a power rule to the variable: \((x^3)^2 = x^{3 \times 2} = x^6\).
The combination of these results gives a final simplified expression of \(64x^6\). These steps clearly break down the process into manageable parts, ensuring that whether numbers or variables are involved, the approach remains consistent and logical.

Basics of Exponential Notation

Exponential notation is a mathematical way to represent numbers as a power of another number. This is especially useful for expressing very large or very small numbers concisely.

• An exponential expression generally has a base and an exponent. The base is the main number you are multiplying, and the exponent tells how many times it is used as a factor.
• For example, in the expression \(x^3\), \(x\) is the base and 3 is the exponent, meaning \(x\times x\times x\).

Exponential notation is integral in mathematics because it simplifies complex calculations and expressions.
In our exercise of \((8x^3)^2\), the expression initially appears complex, but using exponential notation and rules like the power of a power rule helps us express it cleanly as \(64x^6\). Thus, understanding exponential notation is crucial for simplifying and solving exponential expressions efficiently.