Problem 4 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ (2 b)^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \(16b^4\).

Step 1: Identify the Base and Exponent

In the expression \((2b)^4\), the base is \(2b\) and the exponent is 4. The expression indicates that the base \(2b\) needs to be raised to the power of 4.

Step 2: Apply the Power to a Product Rule

The Power to a Product Rule states that \((xy)^n = x^n \, y^n\). Here, apply this rule to the base \(2b\) raised to the 4th power. This gives: \((2b)^4 = 2^4 \times b^4\).

Step 3: Simplify the Numerical Exponent

Calculate \(2^4\). Since \(2^4 = 2 \times 2 \times 2 \times 2 = 16\), the expression now becomes \(16 \times b^4\).

Step 4: Write the Simplified Expression

Combine the results from the previous step to write the fully simplified expression as \(16b^4\).

Key Concepts

Power to a Product Rule

The Power to a Product Rule is a vital tool in algebra that simplifies expressions where a product is raised to a power. It's expressed as \((xy)^n = x^n \, y^n\). This rule helps break down complex expressions into simpler forms by applying the exponent to each part of the product separately.

For example, consider the expression \((2b)^4\). Here, the base is a product \(2b\), and the exponent is \(4\). By applying the Power to a Product Rule, you distribute the power of 4 to both \(2\) and \(b\) separately, resulting in \(2^4 \, b^4\).

This rule not only simplifies calculations but also makes it easier to visualize and solve algebraic problems. By breaking down complex expressions into more manageable pieces, students can easily compute large numbers and work with higher powers without the potential confusion of large exponents.

Simplifying Expressions

Simplifying expressions involves reducing complex algebraic expressions to their simplest form. This process often speeds up solving equations and understanding the problem by removing unnecessary complexity.

The original expression \((2b)^4\) simplifies by applying the Power to a Product Rule, resulting in \(2^4 \, b^4\). The next step involves solving \(2^4\). Breaking it down, \(2^4\) equals \(16\) since \(2 \times 2 \times 2 \times 2 = 16\).

• Break down the expression into manageable parts
• Apply algebraic rules like the Power to a Product Rule
• Calculate powers and combine results

Simplifying keeps equations concise and lessens the risk of errors in later calculations. By practicing simplification, students sharpen their algebraic skills and facilitate problem-solving in more complex scenarios.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division.

These expressions form the basic building blocks of algebra. An expression like \((2b)^4\) contains numbers and variables (here \(2\) and \(b\)) as its components. Each element plays a unique role within the expression, defining its behavior and outcome.

Understanding these components aids students in evaluating, simplifying, and factoring expressions across various algebraic contexts. Learning how different parts interact simplifies the resolution of challenges within algebra. Knowing how to manage exponents like in our example, enables students to handle more intricate equations confidently.

• Recognize elements: numbers, variables, and operations
• Simplify using various algebraic rules
• Apply skills to tackle broader mathematical problems

Mastering algebraic expressions is crucial for advancing in mathematics, providing the tools needed to approach high-level topics like calculus and discrete math with ease.