Problem 50 - Beginning and Intermediate Algebra

Question

Simplify each expression. (Remember the order of operations.) See Examples 4 and 5. $$ 2^{3}-6 \cdot 3 $$

Step-by-Step Solution

Answer

The simplified expression is \(-10\).

Step 1: Apply the Order of Operations (PEMDAS/BODMAS)

The order of operations dictates Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In the given expression \(2^{3}-6 \cdot 3\), we identify an exponent and a multiplication operation.

Step 2: Calculate the Exponentiation

First, handle the exponent: \(2^{3}\). This means \(2 \times 2 \times 2\). Calculate this as follows:\[2^{3} = 2 \times 2 \times 2 = 8\]

Step 3: Perform the Multiplication

Next, perform the multiplication operation \(6 \cdot 3\). Calculate this as:\[6 \cdot 3 = 18\]

Step 4: Perform the Subtraction

With the results from the previous steps, replace the expressions: \(8 - 18\). Subtract to simplify the expression:\[8 - 18 = -10\]

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions, it means breaking down a complex math statement into its simplest form. This involves performing the operation inside the expression while keeping the integrity of the mathematical meaning. In the expression \(2^{3} - 6 \cdot 3\), simplifying is essential because the expression contains different operations like exponents and multiplication. To make sure we simplify correctly, we follow a specific order of operations often remembered as PEMDAS/BODMAS. This stands for Parentheses, Exponents, Multiplication and Division, Addition, and Subtraction.

• Begin by identifying all operations in the expression.
• Apply the order of operations to simplify step by step.
• Perform calculations clearly and carefully for each step.

Breaking down expressions into smaller, manageable parts can help solve math problems accurately and easily.

Exponents

Exponents are a way to express repeated multiplication of the same number. In an expression like \(2^{3}\), it tells us to multiply the base, which is 2, by itself three times: \(2 \times 2 \times 2\). The number 3 in the expression is called the exponent and tells us how many times to multiply the base by itself.
Using exponents can make calculations quicker and expressions easier to read. For instance, instead of writing \(2 \times 2 \times 2\), we simply write \(2^{3}\), and this is much neater and understandable.

• Understand the base and exponent: Base is the number being multiplied, and the exponent defines how many times it is used in multiplication.
• Solve exponents accurately by multiplying the required number of times.
• In the expression example, \(2^{3}\) equals 8, simplifying part of our original expression.

Hence, learning how to work with exponents is a fundamental skill in mathematics that simplifies complex multiplication problems.

Multiplication and Subtraction

Understanding multiplication and subtraction helps you tackle expressions more effectively. In the order of operations, multiplication comes before subtraction. This means you need to solve any multiplication in the expression before any subtraction.
For example, in the expression \(6 \cdot 3\), multiplication results in 18, and this part must be calculated before tackling subtraction. Once multiplication is done, proceed to subtraction as in the original expression:

• Complete multiplication before any subtraction.
• Accurate arithmetic ensures correct results.
• For \(8 - 18\), perform subtraction to simplify it to \(-10\).

Each of these operations requires careful attention to produce accurate results. Whether you're calculating budget or solving algebra, mastering these techniques forms the foundation for successful math problem solving.