Problem 52 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \(-81a^4b^4\).

Step 1: Analyze the Expression

We have the expression \(-(3ab)^4\). This means we need to raise the entire expression \((3ab)\) to the fourth power and then multiply by \(-1\).

Step 2: Distribute the Exponent

Apply the exponent to each factor inside the parentheses. Calculate \((3)^4\), \(a^4\), and \(b^4\) separately. The expression becomes \(-1 \times (3^4) \times (a^4) \times (b^4)\).

Step 3: Calculate the Powers

Compute each part: \(3^4=81\), so the expression becomes \(-1 \times 81 \times a^4 \times b^4\).

Step 4: Apply Multiplication

Finally, multiply the results: \(-1 \times 81 = -81\), therefore the expression simplifies to \(-81a^4b^4\).

Key Concepts

Understanding Monomials

A monomial in algebra is a single term expression that can include numbers, variables, or the product of numbers and variables. Examples of monomials can be just numbers like 7, variables like \( x \), or a combination like \( 3ab \). They do not contain addition or subtraction. Monomials are the building blocks of more complex algebraic expressions.
Monomials help set the stage for performing operations like multiplication and raising to a power, as in our exercise. When working with monomials, ensure you deal correctly with their coefficients (the number parts) and variables for accurate results.

Explaining the Power of a Product Property

The power of a product property is a key concept in exponents. It states that when two or more factors are multiplied together and raised to an exponent, the exponent applies to each factor individually.

• For example, \((3ab)^4\) translates to \(3^4 imes a^4 imes b^4\).
• This property allows simplification of more complex expressions, making calculations more manageable.

Understanding this property is essential when simplifying expressions that involve multiple factors. It ensures that each part of the monomial is treated appropriately, leading to accurate results.

Importance of the Negative Sign in Algebra

An often overlooked yet critical aspect of algebra is the treatment of negative signs. When an expression like \( -(3ab)^4 \) is presented, the negative sign in front affects the entire expression.

• The negative sign multiplies through after the expression is simplified.
• In the given exercise, once \( (3ab)^4 \) is calculated, the result is then multiplied by \(-1\)

This results in the final expression being negative. Ignoring the negative sign can lead to significant errors in algebraic computations, changing the expression's value entirely.

What Are Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition, subtraction, multiplication, and division). They can range from simple monomials to complex polynomials.
In the context of our problem, the expression is \(-(3ab)^4\), which involves both multiplication and exponentiation.

• An understanding of algebraic expressions allows one to manipulate them using algebraic rules and properties like distribution and exponent rules.
• They form the basis of much of the work in algebra, translating real-world problems into a form that can be analyzed and solved mathematically.

Simplifying Expressions Explained

Simplifying expressions in algebra involves making them easier to work with by combining like terms, applying exponent rules, and carrying out arithmetic operations. In our given exercise, simplification entails:

• Calculating \(3^4 = 81 \).
• Multiplying this result by the coefficients of the variables \( a^4 \) and \( b^4 \).
• Lastly, applying the negative sign at the end to get the simplified expression \(-81a^4b^4\).

Simplification allows one to deal with neat and concise results, making further mathematical operations or evaluations more straightforward. It also provides an opportunity to check for understanding of algebraic operations and properties.