Problem 100 Multiply. $$ 0.5 u^{5}\left(... [FREE SOLUTION]

Chapter 5: Problem 100

Multiply. $$ 0.5 u^{5}\left(0.4 u^{6}-0.5 u^{3}\right) $$

Short Answer

Expert verified

The result of multiplying is $0.2 u^{11} - 0.25 u^{8}$.

Step by step solution

Distribute the Monomial

You are given the expression $0.5 u^5(0.4 u^6 - 0.5 u^3)$. The first step is to distribute the monomial $0.5 u^5$ to each term inside the parentheses. This means you'll multiply $0.5 u^5$ by both $0.4 u^6$ and $-0.5 u^3$.

Multiply the First Pair of Terms

Multiply $0.5 u^5$ by $0.4 u^6$:- Coefficients: $0.5 \times 0.4 = 0.2$- Variables: $u^5 \times u^6 = u^{5+6} = u^{11}$Thus, the product is $0.2 u^{11}$.

Multiply the Second Pair of Terms

Multiply $0.5 u^5$ by $-0.5 u^3$:- Coefficients: $0.5 \times -0.5 = -0.25$- Variables: $u^5 \times u^3 = u^{5+3} = u^{8}$Thus, the product is $-0.25 u^{8}$.

Combine the Results

Combine the products from the multiplication steps:$0.2 u^{11} - 0.25 u^{8}$. This is your final expression after multiplying.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property

The Distributive Property is an important algebraic rule that you've likely encountered in math class. It's a way of removing parentheses in an expression by distributing a term across the terms inside the parentheses.
In the expression given:

$0.5 u^5(0.4 u^6 - 0.5 u^3)$

we use the Distributive Property to multiply the monomial, $0.5 u^5$, by each term inside the parentheses. This means:

Multiplying $0.5 u^5$ by $0.4 u^6$
Multiplying $0.5 u^5$ by $-0.5 u^3$

These individual multiplications then replace the original expression, removing the parentheses completely. This simplifies our calculation and brings us closer to solving the algebra problem. It's a core tool you will regularly use when working with expressions.

Exponent Rules

Understanding Exponent Rules allows you to work with terms that include powers of the same base. These rules make it simpler to multiply variables with similar bases by focusing only on the exponents.
Here's a quick rundown:

When multiplying two terms that have the same base, you add their exponents. For example, $u^5 \times u^6 = u^{5+6} = u^{11}$.

In our problem, you apply this rule twice:

For $u^5 \times u^6$ to get $u^{11}$
For $u^5 \times u^3$ to get $u^{8}$

These steps highlight how exponent rules operate to make expressions more manageable during calculations. It helps condense longer polynomial expressions into simpler, easier-to-read terms.

Monomial Multiplication

Monomial Multiplication is straightforward but requires a focus both on multiplying coefficients and combining variables.

When multiplying monomials, first look at the numerical coefficients. Multiply these together.
- Example: $0.5 \times 0.4 = 0.2$ and $0.5 \times -0.5 = -0.25$.
Multiply the variables by adding their exponents if they have the same base.
- Example: $u^5 \times u^6 = u^{11}$ and $u^5 \times u^3 = u^{8}$.

Bringing these products together forms a simplified expression without parentheses:

Resulting Expression: $0.2 u^{11} - 0.25 u^{8}$

Monomial multiplication is crucial for mastering polynomial operations as it forms the foundation for more complex operations later.

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91影视

Multiply. $$ 0.5 u^{5}\left(0.4 u^{6}-0.5 u^{3}\right) $$

Short Answer

Step by step solution

Distribute the Monomial

Multiply the First Pair of Terms

Multiply the Second Pair of Terms

Combine the Results

Key Concepts

Distributive Property

Exponent Rules

Monomial Multiplication

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