Problem 56 - Algebra for College Students

Question

Raise each monomial to the indicated power. $$\left(-4 a b c^{4}\right)^{3}$$

Step-by-Step Solution

Answer

The expression is \(-64a^3b^3c^{12}\).

Step 1: Understanding the Problem

We need to raise the monomial \(-4abc^4\) to the third power. This means we will multiply the monomial by itself three times.

Step 2: Apply Exponentiation to Each Component

Each part of the monomial \(-4\), \(a\), \(b\), and \(c^4\) should be raised to the 3rd power separately. Begin with the coefficient and variables: \((-4)^3\), \(a^3\), \(b^3\), and \((c^4)^3\).

Step 3: Calculate the Powers

Calculate each power: \((-4)^3 = -4 \times -4 \times -4 = -64\); \(a^3 = a^3\); \(b^3 = b^3\); and by using the power of a power property \((c^4)^3 = c^{4 \times 3} = c^{12}\).

Step 4: Write the Final Expression

Combine the results from the previous step: the expression becomes \(-64a^3b^3c^{12}\).

Key Concepts

Monomials

A monomial is a mathematical expression consisting of a single term. This term can be a constant, a variable, or a product of constants and variables. For example, in the expression \(-4abc^4\), each part represents a component of the monomial. Here, \(-4\) is a constant coefficient, while \(a\), \(b\), and \(c^4\) are variables with exponents.
Monomials are building blocks in algebra, serving as essential elements for constructing more complex expressions like polynomials. Understanding how to manipulate monomials—such as raising them to powers—is crucial in simplifying and solving algebraic expressions.
Some key points about monomials include:

• They contain no addition or subtraction signs within the expression.
• They are typically seen in the form of a product of numbers and/or variables.
• Multiplying monomials involves adding the exponents of identical bases.

Power of a Power Property

The power of a power property is a fundamental rule in exponentiation. It states that if you raise a power to another power, you can multiply the exponents. For example, in our problem, we see the portion \((c^4)^3\).
To solve this, we multiply the exponents: \(4 \times 3\), resulting in \(c^{12}\). This method simplifies the computation and helps efficiently manage expressions with nested exponents.
Remember these key points about the power of a power property:

• Only apply the property when both the base and the power are raised to another power.
• The base remains unchanged, while the exponents are multiplied.
• This property is applicable for variables and numerical expressions alike.

Employing this property correctly enhances your problem-solving efficiency when working with exponential expressions.

Polynomial Mathematics

Polynomial mathematics involves expressions that consist of multiple monomials, unlike the single-term monomials we started with.
Polynomials can have various degrees and can be as simple as a monomial or as complex as a multi-term expression. They are crucial in mathematics because they model a wide range of real-world phenomena, and learning how to manipulate them effectively is essential.
When dealing with polynomials, keep these points in mind:

• Each term in a polynomial is a monomial, making understanding monomials foundational.
• Operations such as addition, subtraction, multiplication, and division can be performed between polynomials.
• The degree of a polynomial is determined by the term with the greatest sum of exponents.

In our exercise, by raising the monomial to a power, the result becomes part of the broader topic of polynomials, showcasing how these basic building blocks can be combined to form more complex expressions.