Problem 55 - A Graphical Approach to College Algebra

Question

Find each product. Assume that all variables represent positive real numbers. $$-4 k\left(k^{7 / 3}-6 k^{1 / 3}\right)$$

Step-by-Step Solution

Answer

-4k^{10/3} + 24k^{4/3}

Step 1: Distribute the term outside the parentheses

Multiply the term outside the parentheses, \(-4k\), by each term inside the parentheses, \(k^{7/3}\) and \(-6k^{1/3}\), separately.

Step 2: Multiply \\(-4k\\) by \\(k^{7/3}\\)

The expression becomes \(-4k imes k^{7/3} = -4k^{1 + 7/3}\). Add the exponents: \(1 + 7/3 = 3/3 + 7/3 = 10/3\). This results in \(-4k^{10/3}\).

Step 3: Multiply \\(-4k\\) by \\(-6k^{1/3}\\)

The expression becomes \(-4k imes -6k^{1/3} = 24k^{1 + 1/3}\). Here, a negative times a negative is positive. Add the exponents: \(1 + 1/3 = 3/3 + 1/3 = 4/3\). This results in \(+24k^{4/3}\).

Step 4: Combine the resulting terms

The terms \(-4k^{10/3}\) and \(+24k^{4/3}\) are combined to form the final expression: \(-4k^{10/3} + 24k^{4/3}\).

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when dealing with polynomial multiplication. In this exercise, we encounter expressions like \(k^{7/3}\) and \(k^{1/3}\). Exponents are shorthand for repeated multiplication, and they follow specific rules which simplify the multiplication process.

• Product of Powers Rule: When multiplying like bases, you can add the exponents together. For example, when you have \(k^a \times k^b\), the result is \(k^{a+b}\). This rule was used when multiplying \(-4k \times k^{7/3}\), resulting in \(k^{10/3}\).
• Power of a Power Rule: Although not directly used here, it's a good reminder that when taking a power of a power, you multiply the exponents. For instance, \((k^a)^b = k^{a \cdot b}\).

By mastering these exponent rules, you can efficiently handle complex expressions in algebra, just by simplifying step-by-step.

Distributive Property

The distributive property is a fundamental principle in algebra, especially in polynomial multiplication. It allows us to "distribute" or multiply a single term by each term within a parenthesis in an expression.
In this particular exercise, the expression \(-4k(k^{7/3} - 6k^{1/3})\) requires distributing the \(-4k\) across each term inside the parenthesis:

• First, \(-4k\) is multiplied by \(k^{7/3}\).
• Then, \(-4k\) is applied to \(-6k^{1/3}\), keeping in mind that a negative times a negative gives a positive result.

Using the distributive property, we systematically break down the problem, turning a complex expression into simpler, manageable parts. This property is not only useful in algebraic expressions but also in mental math and basic arithmetic.

Algebraic Expressions

At the heart of algebra are expressions like \(-4k(k^{7/3} - 6k^{1/3})\), which consist of variables, numbers, and operations. Algebraic expressions help us model real-world scenarios and solve mathematical problems.

• Terms: Each part of an expression that is added or subtracted is called a term. In the expression, \(-4k^{10/3}\) and \(+24k^{4/3}\) are terms.
• Coefficients: These are the numerical factors of terms. In our problem, \(-4\) and \(+24\) are coefficients.
• Variables and Exponents: The letters (e.g., \(k\)) in expressions signify variables, which can change. Exponents show the power a variable is raised to, indicating repeated multiplication.

Grasping these components of algebraic expressions enables you to decode and work through algebra problems with confidence, moving beyond simple arithmetic to more abstract reasoning.