Problem 81 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ 20 \sqrt[3]{b^{4}}-4 \sqrt[3]{b} $$

Step-by-Step Solution

Answer

The simplified expression is \(4\sqrt[3]{b}(5b - 1)\).

Step 1: Identify Like Terms

In the expression \(20 \sqrt[3]{b^4} - 4 \sqrt[3]{b}\), notice that both terms contain cube roots of the form \(\sqrt[3]{b^n}\). This means we can try to factor a common term.

Step 2: Find the Common Factor

Identify the greatest common factor from the two terms. The common factor here is \(\sqrt[3]{b}\), as \( b^4 \) can be expressed as \( b^3 \cdot b \) and thus \( \sqrt[3]{b^4} = b\cdot\sqrt[3]{b}\).

Step 3: Factor Out the Common Factor

Factor \(\sqrt[3]{b}\) out of the expression. This simplifies the expression to: \(\sqrt[3]{b}(20b - 4)\).

Step 4: Simplify Inside the Parentheses

Simplify the expression inside the parentheses. Calculate \(20b - 4\) to get \(4(5b - 1)\).

Step 5: Final Expression

The fully simplified expression is \(4\sqrt[3]{b}(5b - 1)\).

Key Concepts

cube roots

Cube roots are an important concept when dealing with expressions that contain terms of the form \(\sqrt[3]{x}\). The cube root of a number \(x\) is a value that, when cubed, gives \(x\) back. This is similar to square roots, but involves the power of three. For example, if \(x = 8\), then \(\sqrt[3]{8} = 2\) because \(2^3 = 8\).

• Cube roots help in simplifying expressions like \(\sqrt[3]{b^4}\) by breaking them down into more manageable parts.
• They allow us to see hidden common factors that can be factored out, making expressions easier to work with.
• When solving, always remember that the cube root operator is applied to the entire number or expression inside the cube root symbol.

Techniques involving cube roots often play a role in factoring, as they allow us to transform complex expressions into simpler polynomial forms. This makes further algebraic manipulation much more straightforward.

factoring

Factoring is the process of breaking down complex expressions into simpler, multi-part forms that, when multiplied together, reproduce the original expression. In our expression \(20 \sqrt[3]{b^4} - 4 \sqrt[3]{b}\), factoring is used to simplify. Let's look at how this works:

• Identify common factors in all terms of an expression. This means looking for variables, constants, or cube roots they share.
• In the given problem, we identified \(\sqrt[3]{b}\) as a common factor in both terms.
• Factoring out this common term, you streamline the process to get expressions like \(\sqrt[3]{b}(20b - 4)\).
• Once factored, these terms inside the parentheses can sometimes be further simplified.

Effective factoring requires recognizing patterns and commonalities within the expression structure, helping to reduce complexity and solve problems more efficiently.

polynomials

Polynomials are expressions composed of variables and coefficients, using operations like addition, subtraction, and multiplication, but never division by a variable. Each term is a product of a constant and a power of a variable. For instance, \(20b - 4\) and \(5b - 1\) from our final expression are simple polynomials. Key characteristics:

• A polynomial's degree is determined by the highest power of its variable.
• Polynomials can be simplified, expanded, and factored, similar to other algebraic expressions.
• They can represent continuous functions that model real-world scenarios, making them indispensable in mathematics.

In dealing with the simplified expression \(4\sqrt[3]{b}(5b - 1)\), knowing how to handle polynomials helps streamline problems that involve factoring and simplifying expressions.

algebraic expressions

Algebraic expressions consist of variables, constants, and various operators. They form the backbone of algebra, allowing representations of mathematical relationships and operations. In our original problem, the expression \(20 \sqrt[3]{b^4} - 4 \sqrt[3]{b}\) is an algebraic expression that undergoes simplification.Key aspects of algebraic expressions include:

• Using variables like \(b\) to represent unknown values or quantities.
• Combining terms through operations: addition, subtraction, multiplication, and division.
• Using rules of simplification, such as factoring, to rewrite expressions in simpler forms for easier manipulation.

The simplification process involves rewriting expressions to make them more efficient to work with or to reveal particular characteristics, like common factors in the expression, ultimately leading to a final concise algebraic form.