Problem 19 Multiply. $$2 a^{2} b\left(5 a... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

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Chapter 10: Problem 19

Multiply. $$2 a^{2} b\left(5 a^{2} b+4 a b\right)$$

Short Answer

Expert verified

The final expression is $10a^{4}b^2 + 8a^{3}b^2$.

Step by step solution

- Distribute the First Term

Multiply the term outside the parentheses by each term inside the parentheses individually. Start with the first term: $2a^2b \times 5a^2b$.

- Simplify the First Product

Combine the like terms and exponents for the first product: $2a^2b \times 5a^2b = 10a^{4}b^2$.

- Distribute to the Second Term

Next, multiply the term outside the parentheses by the second term inside the parentheses: $2a^2b \times 4ab$.

- Simplify the Second Product

Combine the like terms and exponents for the second product: $2a^2b \times 4ab = 8a^{3}b^2$.

- Combine the Results

Add the two products obtained from the distribution: $10a^{4}b^2 + 8a^{3}b^2$.

- Final Simplification

Write the final expression after combining the distributed terms: $10a^{4}b^2 + 8a^{3}b^2$.

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Key Concepts

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

distributive property

When we discuss polynomial multiplication, the distributive property is key. It states that you need to multiply a single term with each term inside the parentheses. Think about it like distributing goods to people - each person gets their share. In the exercise ${{2a^2b(5a^2b+4ab)}}$, we start by distributing ${{2a^2b}}$ to both ${{5a^2b}}$ and ${{4ab}}.$

combining like terms

Once we've used the distributive property, like combining ${{2a^2b \times 5a^2b = 10a^4b^2}}$, we need to tidy up the expression by combining like terms. These terms have the same variables raised to the same power. For example, ${{10a^4b^2}}$ and ${{8a^3b^2}}$ are like terms because they both contain ${b^2}}.$

exponent rules

Exponent rules help us simplify the process of multiplication in polynomials. When multiplying two terms with the same base, you add the exponents: ${a^2 \times a^2 = a^{4}}$. We apply similar rules when distributing, for example, ${(2a^2b) \times (5a^2b) = 10a^4b^2}}.$ Just remember to add the powers.

simplifying expressions

To wrap things up, we simplify the expression. Starting with the distributive property like in our exercise and moving through combining like terms and exponent rules, we boil it down to a cleaner form. After distributing and combining, our expression became ${{10a^4b^2 + 8a^3b^2}}$. This step ensures clarity in the final answer.

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91影视

Multiply. $$2 a^{2} b\left(5 a^{2} b+4 a b\right)$$

Short Answer

Step by step solution

- Distribute the First Term

- Simplify the First Product

- Distribute to the Second Term

- Simplify the Second Product

- Combine the Results

- Final Simplification

Key Concepts

distributive property

combining like terms

exponent rules

simplifying expressions

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