Problem 46 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ (6-3 w)\left(-w^{2}\right) $$

Step-by-Step Solution

Answer

The expression \((6 - 3w)(-w^2)\) without parentheses, applying the distributive property, is -6w^2 + 3w^3.

Step 1: Understand the Problem & Identify the Distributive Property

The Distributive Property is an algebra property which is used to multiply a single term and two or more terms inside a set of parentheses. In this case, the distributive property is \(a(b + c) = ab + ac\). We are asked to remove the parentheses from the expression \((6 - 3w)(-w^2)\).

Step 2: Apply the Distributive Property

Apply the distributive property by multiplying both terms in the parentheses by \(-w^2\). This gives us: -6w^2 + 3w^3.

Step 3: Final Result

After multiplying every term correctly, the expression without parentheses is -6w^2 + 3w^3. And remember, always take care of the negative sign when you are distributing.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In algebra, we often encounter expressions enclosed within parentheses that require simplification for further use. Simplification often involves using the distributive property, an essential tool in algebra.

An expression like the provided exercise \( (6-3w)(-w^2) \) includes both constants and variables. Constants are the numbers on their own (in this case, 6), and the variables are represented by letters which can hold various values (in this case, \( w \) and \( w^2 \) ). The goal of the distributive property is to eliminate parentheses by distributing the multiplication across each term within the parentheses, which fundamentally alters the expression's structure.

Multiplying Polynomials

Multiplying polynomials is a foundational skill in algebra that involves distributing each term of one polynomial across each term of another. When facing an expression like \( (6-3w)(-w^2) \), we are looking at the product of a binomial and a monomial. A binomial is an algebraic expression containing two terms (such as \( 6-3w \)), while a monomial has only one term (like \( -w^2 \) in this case).

To multiply these, you apply the distributive property strategically:

• Firstly, multiply \( -w^2 \) by 6, which gives \( -6w^2 \).
• Then, multiply \( -w^2 \) by \( -3w \) which yields \( 3w^3 \).

Putting these together, we arrive at the final, simplified expression \( -6w^2 + 3w^3 \) without the parentheses.

Negative Exponents

Negative exponents are often a source of confusion but bear a simple meaning. Any nonzero number raised to a negative exponent is equivalent to its reciprocal raised to the corresponding positive exponent. For instance, \( a^{-n} = \frac{1}{a^n} \), assuming \( a \) is not zero.

In the given exercise, \( -w^2 \) isn't a negative exponent by itself since \( w^2 \) stands for \( w \) times \( w \). However, if we had an expression like \( w^{-2} \) it would translate to \( \frac{1}{w^2} \) following the concept of negative exponents. Understanding this concept is crucial when you are simplifying algebraic expressions that contain negative exponents. Always remember that a negative sign in front of an exponent and a negative exponent are two different things.