Problem 11 Find each product. $$ -q^{3}... [FREE SOLUTION]

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Chapter 5: Problem 11

Find each product. $$ -q^{3}(2+3 q) $$

Short Answer

Expert verified

-2q^3 - 3q^4

Step by step solution

- Distribute the outer term

Start by distributing the outer term $-q^3 $ to both terms inside the parentheses $2 + 3q$.

- Multiply the first term

Multiply $-q^3$ by the first term $2$: $-q^3 \cdot 2 = -2q^3$.

- Multiply the second term

Multiply $-q^3$ by the second term $3q$: $-q^3 \cdot 3q = -3q^4$.

- Write the final expression

Combine the results from Steps 2 and 3 to get the final expression: $-2q^3 - 3q^4$.

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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 a fundamental concept in algebra. It allows us to multiply a single term by each term within a set of parentheses. This property is written as: $a(b + c) = ab + ac$. This means you distribute the outer term to every term inside the parentheses.

In our example, we used the distributive property to simplify $-q^3(2 + 3q)$. We multiplied $-q^3$ by each term within the parentheses, $2$ and $3q$, separately.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. In algebra, expressions can be simplified using various properties and rules.

In the given exercise, the expression $-q^3(2 + 3q)$ consists of:

A coefficient (-1) paired with a variable raised to a power (q^3).
Another term within the parentheses (2 + 3q), which is also an algebraic expression.

We simplify the entire expression by applying the distributive property, thus simplifying it step by step until we have a final form without parentheses.

Exponent Rules

Exponents are used to express repeated multiplication of a base number. The basic exponent rules help in simplifying expressions with powers.

Here are some crucial exponent rules:

Product of Powers: $a^m \times a^n = a^{m+n}$.
Power of a Power: $(a^m)^n = a^{m \times n}$.
Power of a Product: $(ab)^n = a^n \times b^n$.

In our example, we used the product of powers rule. When we multiplied $-q^3dot 3q$, exponents added up: $-q^3 \times 3q = -3q^{3+1} = -3q^4$.

Coefficients

Coefficients are the numerical factors in terms of an algebraic expression. They multiply a variable or a product of variables.

In the expression $-q^3(2 + 3q)$, the coefficients are:

-1 in $-q^3$.
2 in $2$.
3 in $3q$.

The coefficients help determine the resulting product when we apply the distributive property. As shown, multiplying $-q^3 \times 2$ gives $-2q^3$, where -2 is the coefficient. Similarly, multiplying $-q^3 \times 3q$ results in $-3q^4$, with -3 being the coefficient.

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91影视

Find each product. $$ -q^{3}(2+3 q) $$

Short Answer

Step by step solution

- Distribute the outer term

- Multiply the first term

- Multiply the second term

- Write the final expression

Key Concepts

Distributive Property

Algebraic Expressions

Exponent Rules

Coefficients

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