Problem 57 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \left(2 v^{3}-8\right)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \(4v^6 - 32v^3 + 64\).

Step 1: Understanding the Expression

We are given the expression \( (2v^3 - 8)^2 \). Our goal is to expand this expression using the formula for the square of a binomial: \( (a - b)^2 = a^2 - 2ab + b^2 \), where \(a\) is \(2v^3\) and \(b\) is \(8\).

Step 2: Calculating \( a^2 \)

First, we calculate \( a^2 \), which is \((2v^3)^2\). Applying the power rule for exponents, we get:\[(2v^3)^2 = 4v^6\].

Step 3: Calculating \( b^2 \)

Next, we calculate \( b^2 \), which is \(8^2\). This simplifies to:\[8^2 = 64\].

Step 4: Calculating \( -2ab \)

Now, we calculate the term \(-2ab\), where \(a = 2v^3\) and \(b = 8\). This term becomes:\[-2(2v^3)(8) = -32v^3\].

Step 5: Combining the Terms

Now combine all the terms from the previous steps to fully expand the original expression:\[(2v^3 - 8)^2 = 4v^6 - 32v^3 + 64\].

Key Concepts

Binomial Expansion

Binomial expansion is an important algebraic tool that helps us expand expressions that are squared or have higher powers. When we talk about a binomial, we mean an expression with two terms, such as \(a + b\) or \(a - b\). To expand such expressions effectively, we rely on certain algebraic formulas. For squaring a binomial, we use:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a - b)^2 = a^2 - 2ab + b^2\)

These formulas allow us to break down the expansion process into smaller, manageable parts. In our example, the expression \((2v^3 - 8)^2\) is a clear case of a binomial squared. We apply the formula for \((a - b)^2\) where \(a = 2v^3\) and \(b = 8\). Each term from the formula helps us expand the original expression step by step, making calculations easier.

Exponents

Exponents represent the number of times a number, called the base, is multiplied by itself. This is a fundamental concept in both algebra and arithmetic. To compute powers accurately, we must understand the rules governing exponents, such as:

• \(a^m \times a^n = a^{m+n}\)
• \((a^m)^n = a^{m \cdot n}\)
• \((ab)^n = a^n \cdot b^n\)

In our exercise, calculating \((2v^3)^2\) involves multiplying the base by itself. Applying the rule \((a^m)^n = a^{m \cdot n}\), we simplify it to \(2^2 \cdot v^{3 \cdot 2}\), which results in \(4v^6\). Understanding these principles allows us to handle powers easily, enhancing our ability to solve various algebraic problems efficiently.

Polynomial Expansion

Polynomial expansion is a technique used to express a polynomial raised to a power as a sum of terms. In this case, we focus on expanding the expression \((2v^3 - 8)^2\) using the binomial expansion technique. Polynomial expressions consist of terms with coefficients and variables raised to powers. When expanding \((2v^3 - 8)^2\), we convert it into a polynomial consisting of separate terms. Our steps included calculating \(a^2\), \(b^2\), and \(-2ab\):

• \(a^2 = (2v^3)^2 = 4v^6\)
• \(b^2 = 8^2 = 64\)
• \(-2ab = -2(2v^3)(8) = -32v^3\)

By evaluating each component and combining them, we achieve a complete polynomial expansion: \(4v^6 - 32v^3 + 64\). Mastery over these steps provides confidence in dealing with various polynomial expressions, paving the way for greater algebraic proficiency.