Problem 40 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ \left(4 x^{6}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \( 16x^{12} \).

Step 1: Understand the Problem

We're given the expression \( (4x^6)^2 \). We need to simplify this expression using the power rule and the power of a product rule.

Step 2: Apply the Power of a Product Rule

According to the power of a product rule, \( (ab)^n = a^n \cdot b^n \). Here, \( a = 4 \) and \( b = x^6 \). So, \( (4x^6)^2 = 4^2 \cdot (x^6)^2 \).

Step 3: Simplify Using the Power Rule

Now, apply the power rule individually to each base. The power rule states that \( (x^m)^n = x^{m\cdot n} \). So, simplify: \( 4^2 = 16 \) and \( (x^6)^2 = x^{12} \).

Step 4: Combine the Results

Combine the results from Step 3. The expression becomes \( 16x^{12} \).

Key Concepts

Power of a Product Rule

When dealing with expressions inside parentheses that are raised to a power, we use the Power of a Product Rule. This rule, stated mathematically as \((ab)^n = a^n \cdot b^n\), helps simplify expressions by distributing the external exponent across each factor inside the expression.

For instance, in the expression \((4x^6)^2\), the factors inside the parenthesis are 4 and \(x^6\). Using the rule, the expression becomes \(4^2 \cdot (x^6)^2\). This step reduces complex expressions into simpler parts that are easier to manage independently. Applying this rule systematically ensures correct simplification of algebraic expressions across various contexts.

Exponentiation

Exponentiation is a fundamental operation in mathematics that involves raising a number or expression to a power. It tells us how many times to multiply a base by itself.

In the exercise, we deal with two main instances of exponentiation:\(4^2\) and \((x^6)^2\). Each instance follows the rules of exponentiation where the base (number or variable) is multiplied by itself as many times as indicated by the exponent.

Key points about exponentiation include:

• The base can be any number or expression.
• The exponent indicates the number of times the base is used as a factor.
• Exponentiation can apply to both numerical and algebraic expressions.

Understanding and applying the rules of exponentiation correctly allows for accurate simplification and manipulation of complex algebraic expressions.

Simplification of Expressions

Simplification is the process of transforming an expression into its simplest form, often making it easier to understand or evaluate. In algebra, simplifying expressions involves reducing them to their most concise form by applying various mathematical rules and operations.

For the expression \((4x^6)^2\), simplification occurs by using exponent rules to generate a simpler expression—\(16x^{12}\).

Steps to simplify include:

• Applying exponent rules consistently.
• Combining like terms whenever possible.
• Ensuring the expression is presented in a straightforward manner with the least complex notation.

The end goal of simplification is to produce an equivalent expression with less complexity, aiding in further operations or calculations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They represent quantities and their relationships in a general form and are foundational to solving algebraic problems.

In the core exercise, \((4x^6)^2\) is an example of an algebraic expression with products and exponents. These kinds of expressions require manipulation using algebraic rules to simplify or solve.

Important characteristics of algebraic expressions include:

• They can contain constants, variables, and exponents.
• Operations such as addition, subtraction, multiplication, and division are used.
• Understanding the structure and properties of algebraic expressions is vital for algebraic problem-solving.

Mastering the manipulation of algebraic expressions forms the foundation for further study in algebra and calculus.