Problem 172 Multiply. $$9 r^{3}\left(r^{2}... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 172

Multiply. $$9 r^{3}\left(r^{2}-3 r+5\right)$$

Short Answer

Expert verified

The result is $9r^5 - 27r^4 + 45r^3$.

Step by step solution

Distribute the Terms

Distribute the term outside the parentheses, which is $9r^3$, to each term inside the parentheses $(r^2 - 3r + 5)$. This gives $9r^3 \times r^2$, $9r^3 \times -3r$, and $9r^3 \times 5$.

Perform the Multiplications

Now, multiply each term:- For $9r^3 \times r^2$, use the laws of exponents to get $9r^{3+2} = 9r^5$.- For $9r^3 \times -3r$, multiply the coefficients and the variables separately to get $9 \times -3 \times r^{3+1} = -27r^4$.- For $9r^3 \times 5$, again, multiply the coefficient with the constant to get $9 \times 5 \times r^3 = 45r^3$.

Combine the Results

Combine the results from Step 2 to form the final expression: $9r^5 - 27r^4 + 45r^3$.

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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 principle in algebra.
It states that when you have a term outside of parentheses, you must multiply it by each term inside the parentheses.
This property helps simplify multiplication problems and distribute the multiplication over addition or subtraction inside the parentheses.
In the exercise, we started with: $$9r^3\big(r^2 - 3r + 5\big)$$
By applying the distributive property, we multiply each term inside the parentheses by $9r^3$.
This translates into three separate multiplications:

$9r^3 \times r^2$
$9r^3 \times -3r$
$9r^3 \times 5$

A helpful way to remember this is to think of distributing flyers: you need to give one to each person (or term) inside the parentheses.

Laws of Exponents

The laws of exponents provide the rules for performing operations involving powers.
In the context of multiplication, one important rule is:
When multiplying two exponents with the same base, you add the exponents.
Here鈥檚 how it鈥檚 applied in our problem:

$9r^3 \times r^2$ becomes $9r^{3+2} = 9r^5$
$9r^3 \times -3r$ becomes $-27r^{3+1} = -27r^4$

Addition of Exponents

Adding exponents is crucial when multiplying terms with the same base.
It simplifies the expression by reducing multiple terms into a single term with a new exponent.
Take for example

$9r^3 \times r^2$:- Here, the bases are the same (both are $r$).
Therefore, you add the exponents: $3 + 2$, resulting in $9r^5$.
Similarly, $9r^3 \times -3r$ transforms through the addition of the exponents: $3 + 1 = 4$, resulting in $-27r^4$.

The key point is to ensure the bases are the same before adding exponents.
Otherwise, the law of addition of exponents does not apply.
By seamlessly applying this rule, algebraic expressions become much easier to handle.

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91影视

Multiply. $$9 r^{3}\left(r^{2}-3 r+5\right)$$

Short Answer

Step by step solution

Distribute the Terms

Perform the Multiplications

Combine the Results

Key Concepts

Distributive Property

Laws of Exponents

Addition of Exponents

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