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Chapter 9: Problem 15

Multiply the following expressions. $$2 x^{4} y^{5}\left(3 x y^{2}+2 x^{2} y+5\right)$$

Short Answer

Expert verified
The expression multiplies to \(6x^5y^7 + 4x^6y^6 + 10x^4y^5\).

Step by step solution

01

Distribute the Monomial

The exercise involves distributing the monomial \(2x^4y^5\) to each term inside the parentheses: \(3xy^2\), \(2x^2y\), and \(5\). This means multiplying the monomial by each term individually.
02

Multiply with the First Term

Multiply \(2x^4y^5\) with \(3xy^2\). This involves multiplying the coefficients (2 and 3) and adding the exponents of like bases (\(x\) and \(y\)). So, \(2 \cdot 3 = 6\), \(x^4 \cdot x^1 = x^{4+1} = x^5\), and \(y^5 \cdot y^2 = y^{5+2} = y^7\). The result is \(6x^5y^7\).
03

Multiply with the Second Term

Multiply \(2x^4y^5\) with \(2x^2y\). Multiply the coefficients (2 and 2) and add the exponents of like bases: \(x^4 \cdot x^2 = x^{4+2} = x^6\) and \(y^5 \cdot y^1 = y^{5+1} = y^6\). The result is \(4x^6y^6\).
04

Multiply with the Third Term

Multiply \(2x^4y^5\) with \(5\). Multiply the coefficients (2 and 5) without any change to the variable part, which adds no additional exponent. Therefore, the result is \(10x^4y^5\).
05

Combine All Products

Add all the products obtained in the previous steps together to get the final expression: \(6x^5y^7 + 4x^6y^6 + 10x^4y^5\).

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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 powerful tool in algebra that makes polynomial multiplication more manageable. When you see an expression like \(2x^4y^5(3xy^2 + 2x^2y + 5)\), the distributive property allows us to break it down by distributing the term outside the parentheses, \(2x^4y^5\), to each term inside the parentheses.Here's how it works:
  • Multiply the term outside the parenthesis by each term inside, one by one.
  • This results in separate products which are then combined to get the final result.
By applying this property, you essentially turn a complex multiplication into several smaller, easier multiplications. It's important to handle each term carefully, ensuring all components - coefficients and variable parts - are accurately addressed in the multiplication process.
Exponents
Exponents tell us how many times a number, known as the base, is multiplied by itself. In the expression \(2x^4y^5\), the \(x^4\) indicates \(x\) is multiplied by itself 4 times, and \(y^5\) indicates \(y\) is multiplied 5 times.When multiplying terms with exponents:
  • Multiply the coefficients as usual.
  • Add the exponents of like bases.
For example, when multiplying \(x^4 \, \text{and}\, x^2\), add the exponents: \(x^{4+2} = x^6\). Similarly, for \(y^5 \, \text{and}\, y^1\), add the exponents: \(y^{5+1} = y^6\). Adding exponents only works when the bases are identical, otherwise, they cannot be combined this way. Understanding how to work with exponents is crucial as it simplifies the multiplication of polynomial expressions significantly.
Monomials
A monomial is a single term consisting of a product of numbers and variables raised to powers. The expression \(2x^4y^5\) is an example of a monomial, where 2 is the coefficient, and \(x^4y^5\) represents the variable part.Key characteristics of monomials:
  • They contain one term without any addition or subtraction signs.
  • The variables in a monomial can have exponents, including zero or positive integers.
  • Monomials can include any combination of numbers, variables, and their exponents.
When multiplying monomials, each part must be carefully considered. Coefficients are multiplied together, and each variable part is handled individually by applying the rules of exponents. Multiplied correctly, monomials can be combined to form larger polynomial expressions through addition or subtraction, as seen in the exercise's solution.

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Most popular questions from this chapter

Simplify each expression, and write all answers in scientific notation. $$\frac{\left(2 \times 10^{-3}\right)\left(6 \times 10^{-5}\right)}{3 \times 10^{-4}}$$

Simplify the following expressions. \(\left(a^{5} b^{3}\right)^{2}\left(a b^{3}\right)^{4}\left(a^{3} b\right)^{5}\)

Expand and multiply. $$(x-4)^{2}$$

Find each quotient. Write all answers in scientific notation. $$\frac{750,000,000}{250,000}$$

Simplify the following expressions. \(\left(a^{4}\right)^{5} \cdot\left(a^{3}\right)^{6}\)
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