Problem 42 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$-2(3)^{2}-2(-2)^{3}-6(-1)^{5}$$

Step-by-Step Solution

Answer

The simplified expression is 4.

Step 1: Simplify the Exponentials

First, calculate the value of each exponential term separately: 1. \((3)^{2} = 9\) 2. \((-2)^{3} = -8\) 3. \((-1)^{5} = -1\)

Step 2: Multiply Coefficients with Exponentials

Next, multiply each coefficient with the result of their corresponding exponential:1. \(-2 \times 9 = -18\) 2. \(-2 \times -8 = 16\) 3. \(-6 \times -1 = 6\)

Step 3: Combine the Results

Add the results from Step 2:\(-18 + 16 + 6 = 4\)

Key Concepts

Exponential Functions

Exponential functions are vital in math and represent repeated multiplication. When you see an expression like \(a^n\), the number \(a\) is called the base, and \(n\) is the exponent. It tells you how many times to multiply \(a\) by itself.

For example, in the expression \((3)^2\), the base is 3, and the exponent is 2, which means we multiply 3 by itself:

• \((3) \times (3) = 9\)

In another expression, like \((-2)^3\), the base is \(-2\) and the exponent is 3, which means:

• \((-2) \times (-2) \times (-2) = -8\)

You might wonder why the sign changes. This is because odd and even exponents affect negative numbers differently. When an exponent is odd, the result of a negative base remains negative, like \((-2)^3 = -8\). Conversely, if the exponent is even, the result becomes positive, like \((-2)^4 = 16\).

Understanding exponential functions helps simplify expressions accurately.

Step-by-Step Solutions

Breaking down complicated problems into simpler, manageable parts is what step-by-step solutions are all about. Let's revisit our problem:
\(-2(3)^2 - 2(-2)^3 - 6(-1)^5\).

The first step is to resolve the exponentials. It’s crucial to handle one operation at a time for clarity. Here's the process:

• Calculate \((3)^{2}\) to obtain 9.
• Calculate \((-2)^{3}\) to yield -8.
• Calculate \((-1)^{5}\) resulting in -1.

Next, you take the results and multiply them by the coefficients, which are the numbers right in front of these expressions:

• Multiply \(-2\) by \(9\) to get \(-18\).
• Multiply \(-2\) by \(-8\) to get \(16\).
• Multiply \(-6\) by \(-1\) to get \(6\).

The last step is to add all these results together:

• Add \(-18\), \(16\), and \(6\) to get a final result of 4.

This structured approach makes each step clear and manageable.

Multiplying Coefficients

Once you've calculated the exponential values, it's time to deal with the coefficients, which are the numbers placed before the exponential terms. This step requires you to multiply these coefficients with the simplified exponential results. This process combines the two parts into a single numeric value.

For example, consider \(-2(3)^2\). First, calculate the exponential part as explained before, \(3^2 = 9\). Now take \(-2\) and multiply it by 9:

• \(-2 \times 9 = -18\)

The same method applies to the other parts of the expression. For \(-2(-2)^3\), after simplifying, you find \((-2)^3 = -8\). Hence:

• \(-2 \times -8 = 16\)

And for \(-6(-1)^5\), the result is:

• \(-6 \times -1 = 6\)

This multiplication ensures each coefficient interacts properly with its exponential, preparing the expression for the final combination.