Problem 49 - Beginning and Intermediate Algebra

Question

Simplify each expression. (Remember the order of operations.) See Examples 4 and 5. $$ 3^{3}-8 \cdot 9 $$

Step-by-Step Solution

Answer

The expression simplifies to -45.

Step 1: Calculate the Exponent

The expression starts with an exponent. Calculate \(3^3\) which means \(3\) multiplied by itself three times: \(3 \times 3 \times 3 = 27\).

Step 2: Perform Multiplication

Now, perform the multiplication part of the expression. Multiply \(8\) by \(9\) to get \(8 \times 9 = 72\).

Step 3: Subtract to Simplify

Subtract the result from Step 2 from the result in Step 1: \(27 - 72\). This equals \(-45\).

Key Concepts

Exponents

In mathematics, exponents describe a number that is being multiplied by itself a specific number of times. This is a form of repeated multiplication and is sometimes referred to as powers. Understanding how to handle exponents is crucial for simplifying expressions correctly.To calculate the exponent in our problem, we start with the base number, which is 3 in this case. The exponent is 3 also, indicating that the base number (3) is to be multiplied by itself a total of three times:- First, multiply 3 by itself: \( 3 \times 3 = 9 \).- Then, take that result and multiply by 3 once more: \( 9 \times 3 = 27 \).So, \( 3^3 = 27 \). Think of this process like building layers; each additional multiplication by the base number builds on the previous result. Using exponents is a quick way to convey this repetitive multiplication in a tidy format.

Multiplication

Multiplication is one of the basic operations in arithmetic, involving the combining of equal groups. When you see the multiplication symbol (\(\cdot\) or \(\times\)), it signals that you should calculate the product of the numbers.In our example, after calculating the exponent, the next step following the order of operations is multiplication. The numbers involved here are 8 and 9.- We calculate as follows: \( 8 \times 9 = 72 \).Multiplication has a property known as commutativity, meaning that the order of numbers doesn't affect the product. Whether you compute \( 8 \times 9 \) or \( 9 \times 8 \), the result is the same, 72.This property is helpful as it allows flexibility in how you can handle multiplication tasks, especially when rearranging numbers for easier calculation or in solving complex mathematical problems efficiently.

Subtraction

Subtraction is the arithmetic operation of taking one number away from another. Once you've handled exponents and multiplication, subtraction often comes last in expressions due to the order of operations (PEMDAS/BODMAS).Upon reaching the subtraction phase in our problem, you have two numbers from the previous steps:- The result from the exponent calculation: 27.- The result from the multiplication: 72.Now, you subtract these numbers: \( 27 - 72 \).It's important to identify which number is being subtracted from which. Here, you take the smaller number (27) and subtract a larger number (72) from it. This results in a negative number because subtracting a larger number from a smaller one always yields a negative outcome:- \( 27 - 72 = -45 \).Negative numbers can initially be tricky, but they simply represent values less than zero. In practical terms, think of negative 45 as being 45 units below zero on a number line.