Problem 92 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

311

Step 1: Evaluate the exponent

Calculate \( (-8)^{2} \). An exponent of 2 means to multiply the number by itself, so \( (-8)^{2} \) is equal to \( (-8) \times (-8) = 64 \).

Step 2: Multiply by 10

Multiply the result by 10. Thus, \( 64 \times 10 = 640 \).

Step 3: Divide by 2

Divide the result by 2. So, \( 640 \div 2 = 320 \).

Step 4: Subtract 9

Finally, subtract 9 from the result. Therefore, \( 320 - 9 = 311 \).

Key Concepts

Exponents

In mathematics, an exponent refers to the number of times a number, called the base, is multiplied by itself. For example, in the exercise, we see the expression \( (-8)^2 \). Here, the base is -8 and the exponent is 2. This means that we need to multiply -8 by itself: \( (-8) \times (-8)\). Let's break this down:

• First, consider the negative signs. Multiplying two negative numbers results in a positive number.
• Next, perform the multiplication: \(8 \times 8 = 64\).

So, \( (-8)^{2} = 64 \). Understanding exponents is crucial because they indicate how many times to use the base in multiplication.

Multiplication

Multiplication is a basic arithmetic operation where you add a number to itself a specified number of times. After evaluating the exponent, we got the number 64. Now, the next step in the exercise is to multiply 64 by 10: \( 64 \times 10 \). To do so, you can think of it as adding 64 ten times:
\[ 64 + 64 + 64 + 64 + 64 + 64 + 64 + 64 + 64 + 64 = 640 \].
Multiplication is a fundamental operation frequently used in more complex math problems.

Division

Division is another key arithmetic operation in which a number is evenly distributed into groups. In the problem, we need to divide the result we obtained from our multiplication step, 640, by 2: \( 640 \div 2 \). To do this, you can think of it as finding how many times 2 fits into 640:

• First, take 640 and break it down: 640 divided by 2 is essentially asking how many 2's are in 640.
• So, \( 640 \div 2 = 320 \).

Division helps us partition items into specific amounts, which is useful in various mathematical and real-world problems.

Subtraction

Subtraction is the process of removing a number from another number. It's the final operation in our exercise. After dividing 640 by 2, we got 320. The next step is to subtract 9 from 320:
\[ 320 - 9 = 311 \].
Here’s a step-by-step guide to this process:

• Line up the numbers vertically if needed, then subtract each digit starting from the right.
• 320 minus 9 equals 311, as there is no need for borrowing or carrying.

Understanding subtraction is essential because it allows us to determine differences, which is fundamental in various math contexts.