Problem 90 - Elementary Algebra

Question

For exercises 15-100, evaluate. $$ -8^{2} \cdot 10 \div 2-9 $$

Step-by-Step Solution

Answer

-329

Step 1 - Evaluate the exponent

First, evaluate the exponent -8^{2}to get -64.

Step 2 - Multiply by 10

Next, multiply -64 by 10 to get -(64 × 10) = -640

Step 3 - Divide by 2

Now, divide -640 by 2 to get: -640 ÷ 2 = -320

Step 4 - Subtract 9

Finally, subtract 9 from -320, -320 implies - - 9= -329.

Key Concepts

Understanding Exponents

To master the order of operations, one of the first concepts you need to understand is exponents. An exponent represents the number of times a base number is multiplied by itself. For example, in \(8^2\), the base is 8 and the exponent is 2. This means you multiply 8 by itself: 8 \times 8 = 64.
However, be cautious with negative numbers. When an exponent is applied to a negative base, the result may differ depending on the position of the negative sign. In our exercise, the negative sign is outside the exponent: \(-8^2\). This translates to: \(-(8^2)\), which is \(-64\) and not \(64\).
It's crucial to perform exponents before any other operation. In our exercise, evaluating \(-8^2\) gives us \(-64\).

Performing Multiplication

Once you have handled the exponents, the next step in the order of operations is multiplication. Multiplication is simply repeated addition. For instance, \(3 \times 4\) means you add 3 four times: (3 + 3 + 3 + 3 = 12).
In our exercise, after finding \(-8^2 = -64\), we multiply \(-64 \times 10\). This operation is straightforward: \text{times} \text{10} = -640). This step follows the multiplication rule directly after exponentiation. Remember to keep the negative sign as it is.

Executing Division

After multiplication, follow with division. Division is splitting a number into equal parts. For example, \(20 \, \div \,4\) means dividing 20 into 4 equal parts, resulting in 5.
In our problem, after multiplying to get \(-640\), the next step is to divide by 2. That is, \(-640 \, \div \,2\). This gives us: -320. Division comes immediately after multiplication in the sequence of the order of operations. Follow the rule \(\frac{-640}{2} = -320\).
Always ensure you divide the result from multiplication accurately.

Subtraction Explained

Finally, it's time for subtraction, the last operation. Subtraction means taking away one number from another. For instance, \(10 - 3\) leaves you with 7.
To finish our exercise, after division we need to subtract 9 from -320: \(-320 - 9\). Think of it as moving 9 units to the left on the number line from -320, giving \(\text{-329}\).
Remember to pay attention to the signs and the order in which you subtract numbers, as it can change the result entirely.

Consequently, the overall solution to the problem follows the order of operations correctly, leading to -329.