Problem 7 Multiply the following expressio... [FREE SOLUTION]

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Prealgebra

Charles P. McKeague

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 9: Problem 7

Multiply the following expressions. $$4 x^{2} y^{3}\left(3 x^{4}+2 y^{3}\right)$$

Short Answer

Expert verified

The product is $12x^6y^3 + 8x^2y^6$.

Step by step solution

Distribute the first term

Multiply the term $4x^2y^3$ by each term inside the parenthesis separately. First, multiply $4x^2y^3$ by $3x^4$. Use the distributive property: \[4x^2y^3 imes 3x^4 = 12x^{2+4}y^{3} = 12x^{6}y^{3}\]This gives the first term of the resulting expression.

Distribute the second term

Next, multiply $4x^2y^3$ by $2y^3$. Again, use the distributive property: \[4x^2y^3 imes 2y^3 = 8x^{2}y^{3+3} = 8x^{2}y^{6}\]This gives the second term of the resulting expression.

Combine the results of distributive steps

Combine the terms obtained from Steps 1 and 2 to form the full expanded expression: \[12x^6y^3 + 8x^2y^6\] This is the final expanded form of the original expression after multiplication.

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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 algebraic concept. It states that for all real numbers, the expression $a(b + c)$ can be distributed to $ab + ac$. This property helps simplify equations and makes it possible to multiply terms separately before combining them.

In the context of our exercise, the distributive property allows us to multiply the expression $4x^2y^3$ by each term inside the parentheses, $3x^4 + 2y^3$.

First, distribute $4x^2y^3$ to $3x^4$.
Then, distribute $4x^2y^3$ to $2y^3$.

This handy property helps break down complex expressions into simpler components, making it easier to handle polynomial multiplication.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials together. It requires applying the distributive property repeatedly to combine all terms from one polynomial with each term in the other polynomial.

Here's a brief breakdown of the multiplication:

Take each term of the first polynomial.
Multiply this term with each term of the second polynomial individually.
Combine all results into a single polynomial by summing like terms.

In the exercise, we multiplied $4x^2y^3$ with each term of the polynomial within the parentheses $3x^4 + 2y^3$. Doing this methodically leads to an expanded expression $12x^6y^3 + 8x^2y^6$. This approach ensures all parts of the polynomials are accounted for, preventing any term from being left out.

Exponent Rules

Exponent rules are essential for simplifying expressions that involve powers. They help us manage how exponents behave during operations like multiplication. Among the exponent rules, here are two key ones involved in this exercise:

Product of Powers Rule: $x^a \times x^b = x^{a+b}$. When multiplying terms with the same base, you add their exponents.
Power of a Power Rule: If you needed $(x^a)^b$, you would multiply the exponents to get $x^{a\times b}$, though this rule wasn't directly used in the current exercise.

For example, when multiplying $4x^2y^3$ by $3x^4$, we used the Product of Powers Rule:

Combine $x^2$ and $x^4$ to get $x^{2+4} = x^6$.
Similarly, the $y$ terms were combined as $y^{3+3} = y^6$.

These rules simplify mathematical operations involving exponents, reducing errors and easing calculations.

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91影视

Multiply the following expressions. $$4 x^{2} y^{3}\left(3 x^{4}+2 y^{3}\right)$$

Short Answer

Step by step solution

Distribute the first term

Distribute the second term

Combine the results of distributive steps

Key Concepts

Distributive Property

Polynomial Multiplication

Exponent Rules

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