Problem 40 - Elementary Algebra Within Reach

Question

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ (10 t)\left(-4 t^{2}\right) $$

Step-by-Step Solution

Answer

-40t^3

Step 1: Understand the expression

The given expression is \( (10 t)(-4 t^2) \).When we look at this expression we see it is multiplication of two terms: \(10 t\) and \(-4 t^2\). Our task is to simplify it. To do so, we will multiply these two terms together.

Step 2: Multiply

Multiplication is done in two steps: multiplying coefficients (the numbers) and multiplying variables.\n- Multiply the coefficients: 10 and -4. The answer is -40.\n- Multiply the variables: \(t\) and \(t^2\). When multiplying variables with exponents, we add the exponents. Here \(t\) can be written as \(t^1\). So the multiplication will give us \(t^{(1+2)}\), which is \(t^3\).

Step 3: Write down the simplified form

After multiplications, we combine the results for coefficients and variables. So, \(10 t * (-4 t^2)\) simplifies to \(-40t^3\).

Key Concepts

Multiplying Coefficients

When you see an expression like \((10t)(-4t^2)\), start by identifying the coefficients. Coefficients are the numerical parts of terms in algebra. In this case, the numbers you see before the variables are coefficients: 10 and -4.
To multiply these coefficients:

• Multiply the numbers as you normally would: 10 times -4.
• This calculation is straightforward and results in -40.

Remember, multiplying a positive number by a negative number results in a negative number. So, the sign of the result will be negative in this case.

Multiplying Variables

After dealing with the coefficients, focus on the variables themselves. Here, we're looking at \(t\) and \(t^2\). These variables have exponents, which help us understand how many times the variable is used in a multiplication.
To multiply variables:

• Write the variables as they are: \(t\) is actually \(t^1\).
• The expression \(t\cdot t^2\) means you are multiplying \(t^1\) with \(t^2\).

You'll need to take a closer look at the exponents next!

Adding Exponents

One important rule in algebra is that when you multiply terms that have the same base, you add their exponents. Let’s revisit our expression with this in mind: \(t^1\) and \(t^2\).
Here's the process for adding exponents:

• Identify each exponent: \(t^1\) and \(t^2\) show this clearly.
• When these terms are multiplied, their exponents are added: \(1 + 2\).
• Therefore, the multiplication results in \(t^3\).

This rule of adding exponents only applies when the base (in this case, \(t\)) is the same for both terms. Finally, combine the multiplied coefficient \(-40\) with the new expression \(t^3\) to get \(-40t^3\).
This simplified form combines both the arithmetic of coefficients and the laws of exponents into one neat expression!