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Chapter 5: Problem 105

Multiply. $$ (2 a)(-5 a b) $$

Short Answer

Expert verified
-10 a^2 b

Step by step solution

01

Multiply the constants

First, multiply the numerical coefficients from each term. In this case, multiply 2 by -5.\(2 \times -5 = -10\)
02

Multiply the variables

Next, multiply the variables. Start by multiplying the like terms.\(a \times a = a^2\)
03

Combine the results

Now, combine the results of the coefficients and the variables to get the final product.\((-10) a^2 b\)

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Key Concepts

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

multiplying coefficients
When you multiply expressions in algebra, the first step is to handle the numerical parts, also known as coefficients. A coefficient is the number directly in front of a variable. In our example, the coefficients are 2 and -5.

To multiply these coefficients:
  • Simply take \(2 \times -5\) and work it out, which gives \(-10\).
This process sets up the numerical value for the final expression.
A useful tip: Remember to pay attention to the signs (positive or negative). In our case, a positive number multiplied by a negative number always results in a negative product.
multiplying variables
Next, we need to multiply the variables from each term. This helps in creating a new, more simplified term.

In the given example, look at the variables in each part:
  • The first term is \(2a\), which contains the variable \(a\).
  • The second term is \(-5ab\), which contains \(a\) and \(b\).
We multiply the similar variables together:
  • Here, \(a \times a = a^2\).
When you multiply variables, just add their exponents. Since both 'a' have an exponent of 1, their product is \(a^2\). The \ b \ remains unchanged because it doesn't have another \ b \ to pair with in the first term.
combining results in algebra
After multiplying the coefficients and variables, the final step is to combine these results. By putting everything together, we form the complete expression.

From our exercise:
  • We got \(-10\) from multiplying the coefficients.
  • We got \(a^2\) from multiplying the variables 'a'.
There is also a 'b' in the second term. Since it has no pair in the first term, it stays as is. The final combined result is:
  • \(-10a^2b\)
Thus, the multiplication of \( (2a) \times (-5ab) = -10a^2 b\).
Combining all parts correctly is crucial for accurate algebraic expressions.

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Most popular questions from this chapter

To understand how the special product \((a+b)^{2}=a^{2}+2 a b+b^{2}\) can be applied to a purely numerical problem. Evaluate \(35^{2},\) using either traditional paper-and-pencil methods or a calculator.

Perform each division. $$ \frac{5 x^{3}+4 x^{2}+10 x+20}{5 x+5} $$

Use scientific notation to calculate the answer to each problem. In \(2008,\) the U.S. government collected about \(\$ 4013\) per person in personal income taxes. If the population was \(304,000,000,\) how much did the government collect in taxes for \(2008 ?\)

Perform each division using the "long division" process. $$ \frac{3 k^{3}-4 k^{2}-6 k+10}{k-2} $$

Multiply. $$ -3\left(x^{2}+7\right) $$
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