Problem 41 - Algebra 1

Question

Simplify the variable expression. $$-\frac{3}{7}\left(-w^{2}\right)(7 w)$$

Step-by-Step Solution

Answer

\(-\frac{9}{7}w^{3}\)

Step 1: Distribute the -3/7 to both terms in the parenthesis

In this step, we are supposed to multiply \(-\frac{3}{7}\) by \(-w^{2}\) and \(7w\). Doing so we get \(-\frac{3}{7} * -w^{2} = \frac{3}{7}w^{2}\) and \(-\frac{3}{7} * 7w = -3w\).

Step 2: Multiply the two resulting terms

Here we multiply \(\frac{3}{7}w^{2}\) by \(-3w\). Doing so, we get \(\frac{3}{7}w^{2} * -3w = -\frac{9}{7}w^{3}\). Applying the power rule \(a^{m}a^{n} = a^{m+n}\), the total exponent for \(w\) becomes \(2 + 1 = 3\).

Step 3: Simplify the expression

After multiplication, the simplified expression becomes \(-\frac{9}{7}w^{3}\).

Key Concepts

Distributive Property

The distributive property is a very useful tool in mathematics that helps simplify complex expressions. It allows you to multiply a single term by two or more terms inside a set of parentheses. Imagine you have a bag with apples and oranges, if you multiply the number of bags by both, you know how many fruits you have in total. This is very similar in algebra. To use the distributive property effectively, you will multiply each term in the parentheses by the term outside. For example, in our original problem: - Multiply \(-\frac{3}{7}\) by \(-w^{2}\) to get \(\frac{3}{7}w^{2}\).- Then, multiply \(-\frac{3}{7}\) by \(7w\) to get \(-3w\).This property is particularly helpful because it allows us to break down problems into smaller manageable parts. It also ensures we meet the rules of proper mathematical syntax.

Power Rule in Algebra

The power rule in algebra is centered around the idea that when you multiply similar variables with exponents, you simply add the exponents together. It's like climbing steps on a staircase: each step moves you up one level. This can simplify complex multiplication involving variables.Consider multiplying \(w^{2}\) and \(w\). You can use the power rule, which states \(a^{m}a^{n} = a^{m+n}\), to find the total power of \(w\).

• In this scenario: \(w^{2} * w^{1} = w^{2 + 1} = w^{3}\).
• This allows you to combine the terms into one single expression.

Applying the power rule properly helps streamline problems and brings simplicity to what may seem complex at first glance.

Multiplying Fractions

Understanding how to multiply fractions is a key part of solving algebraic expressions with variables. Fractions represent parts of a whole, and multiplying them isn't as daunting as it may seem.When multiplying fractions, the process is straightforward:

• First, multiply the numerators (top numbers) together.
• Next, multiply the denominators (bottom numbers) together.

For instance, in our problem, we multiply \(-\frac{3}{7}\) by \(7w\) and see how the numeric part calculates:

• The -3 and 7 multiply to \(-3 \times 1 = -3\), and 7 (from the denominator) cancels out the 7 from the variable \(7w\).
• This simplifies to \(-3w\).

Creating easier fractions by simplifying can make the math more direct and clearer as you follow through the steps.