Problem 46 Find each product. $$\left(-a ... [FREE SOLUTION]

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Chapter 5: Problem 46

Find each product. $$\left(-a b^{5}\right)\left(-2 a^{2} b\right)$$

Short Answer

Expert verified

2a^3b^6

Step by step solution

- Write out the expression

Copy down the expression given in the problem: $\big(-a b^{5}\big)\big(-2 a^{2} b\big)$

- Multiply the constants

Multiply the numerical coefficients: -1 and -2. $-1 \times -2 = 2$.

- Multiply the variable 'a' terms

Add the exponents for the variable $a$. $a \times a^{2} = a^{1 + 2} = a^{3}$.

- Multiply the variable 'b' terms

Add the exponents for the variable $b$. $b^{5} \times b^{1} = b^{5+1} = b^{6}$.

- Combine all the terms

Putting it all together, we get: $2 a^{3} b^{6}$.

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Key Concepts

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

Exponent Rules

Understanding exponent rules is essential for multiplying algebraic expressions. In this exercise, we need to apply these rules to simplify the terms for each variable.
When multiplying terms with the same base, add the exponents together. For example:
鈥� For the variable 'a': $-a \times a^{2} = a^{1 + 2} = a^{3}$
鈥� For the variable 'b': $b^{5} \times b^{1} = b^{5+1} = b^{6}$
This addition rule holds for any base that appears in both terms being multiplied. It鈥檚 a simplifying process that pools together the powers to create a single term with one exponent.

Variables

Variables are symbols that represent numbers. In algebra, we often use letters like $a$ and $b$ to stand in for unknown values.
In our exercise, we have two types of variables: $a$ and $b$. Each variable can be multiplied separately using the exponent rules explained before.
Consider: $(-a b^{5})(-2 a^{2} b)$
We treat $-a$ and $-2 a^{2}$ separately, focusing on the variable $a$ first:
Combining the powers of $a$: $a \times a^{2} = a^{1 + 2} = a^{3}$
Then handle the variable $b$: $b^{5} \times b = b^{5+1} = b^{6}$
By carefully working with each variable, we can simplify even complex algebraic expressions.

Constants Multiplication

Constants in algebra are numbers without variables. Multiplying constants follows simple arithmetic rules.
In our step-by-step solution, the constants given are \-1\ and \-2\. We multiply these as usual: $-1 \times -2 = 2$
The negative signs cancel out each other, resulting in a positive product.
By multiplying constants early in the process, we simplify the expression quickly, preparing it for further simplification. Combining constants first allows us to handle the variable terms more efficiently in subsequent steps. Together, constants and variables make up the full terms of an expression, and proper handling of each is crucial for correct simplification.

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91影视

Find each product. $$\left(-a b^{5}\right)\left(-2 a^{2} b\right)$$

Short Answer

Step by step solution

- Write out the expression

- Multiply the constants

- Multiply the variable 'a' terms

- Multiply the variable 'b' terms

- Combine all the terms

Key Concepts

Exponent Rules

Variables

Constants Multiplication

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