Problem 61 - Elementary Algebra

Question

For exercises 15-100, evaluate. $$ -3^{2}+8(-2) $$

Step-by-Step Solution

Answer

-25.

Step 1: Evaluate the exponent

First, calculate the exponent part of the expression. 2. Remember that the exponentiation is performed before multiplication due to the order of operations (PEMDAS/BODMAS). 3. Evaluate 3^2, which is equal to 9. Therefore, the term -3^2 can be written as -9.

Step 2: Multiply 8 by -2

Now, evaluate the multiplication part of the expression. Multiply 8 by -2: 8(-2) = -16.

Step 3: Final Calculation

Combine the results from steps 1 and 2. Thus, the complete expression is -9 + (-16). -9 and -16 are both negative numbers. When adding two negative numbers, add their absolute values and keep the negative sign: 9 + 16 = 25. Therefore, -9 + (-16) = -25.

Key Concepts

Exponentiation

Exponentiation is the process of raising a base number to the power of an exponent. It is represented as \(a^b\), which means that the base number \(a\) is multiplied by itself \(b\) times. This mathematical operation follows strict rules:

• It is always performed before multiplication and addition in an equation (following the order of operations).
• An exponent tells you how many times to use the base as a factor.

In our example, we have \(-3^2\). Here, the base is -3, and the exponent is 2, which means we multiply -3 by itself once.
But remember, exponentiation comes before applying the negative sign due to the order of operations. So, we first compute the exponentiation part: \(3^2 = 9\). Then we apply the negative sign, resulting in -9.

Multiplication

Multiplication involves combining equal groups. When multiplying a positive number by a negative number, the result is always negative:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative

For our exercise, we perform the multiplication 8 × -2.
According to the rules, the result will be negative since one number is positive and the other is negative.
Hence, 8 × -2 equals -16.

Addition of Negative Numbers

Adding negative numbers might seem tricky initially, but there are simple rules to follow. When adding two negative numbers, you:

• Ignore the negative signs and add their absolute values.
• Put the negative sign in front of the result.

In our expression, we combine -9 (result from exponentiation) and -16 (result from multiplication):

• Step 1: Ignore signs and add absolute values: 9 + 16 = 25
• Step 2: Add the negative sign to the result: -25

Therefore, -9 + (-16) equals -25.