Problem 102 - Elementary Algebra

Question

For exercises 101-104, evaluate. Use a scientific or a graphing calculator. $$ -8^{2} \cdot 2-9 $$

Step-by-Step Solution

Answer

-137

Step 1 - Evaluate the exponent

Evaluate \(8^2\). Since the exponent has a higher precedence than the negation, this results in \(8^2 = 64\).

Step 2 - Apply the negation

Apply the negation to the result of the exponentiation. That gives us \(-64\).

Step 3 - Perform multiplication

Next, multiply the result by 2: \(-64 \times 2 = -128\).

Step 4 - Perform subtraction

Finally, subtract 9 from the result of the multiplication: \(-128 - 9 = -137\).

Key Concepts

Exponents Explained

Exponentiation is a mathematical operation that raises a number, called the base, to the power of an exponent. For instance, in the expression \( 8^2 \), the number 8 is the base, and 2 is the exponent. Exponentiation tells us to multiply the base by itself as many times as indicated by the exponent. So, \( 8^2 \) is calculated as \( 8 \times 8 = 64 \). This step must be performed first, according to the order of operations (PEMDAS/BODMAS). In our original problem, we computed \( 8^2 = 64 \) before moving on to other operations.

Understanding Negation

Negation is the process of changing the sign of a number. When negating a number, you simply flip its sign from positive to negative or vice versa. This is especially important in calculating expressions correctly. In our example, after determining that \( 8^2 = 64 \), we apply the negation to get \( -64 \). Note that negation has a lower precedence than exponentiation, meaning if both appear in an expression, you perform the exponent first and then apply the negation.

Multiplication Is Key

Multiplication is a fundamental arithmetic operation that combines groups of equal sizes. In an expression, it is crucial to follow the order of operations by performing multiplication before addition or subtraction. For our problem, after applying negation and getting \( -64 \), we multiply by 2. This means calculating: \( -64 \times 2 \). Multiplying negative and positive numbers preserves the negative sign: \( -64 \times 2 = -128 \). It is essential to be careful with the signs to avoid mistakes.

Simplifying with Subtraction

Subtraction is the operation of finding the difference between numbers. It's one of the last steps to perform in an arithmetic expression following the order of operations rules. As the final step in our exercise, we subtract 9 from the result of our multiplication. Starting with \( -128 \), we calculate: \( -128 - 9 = -137 \). Remember that subtracting a positive number from a negative number effectively makes the negative number larger in magnitude, leading us to our final answer of \( -137 \).