Problem 112 - Elementary and Intermediate Algebra

Question

a. \(2 \cdot 3^{3}\) b. \((2 \cdot 3)^{3}\)

Step-by-Step Solution

Answer

a. 54 b. 216

Step 1: Understanding the Expression for Part a

The expression given for part a is \(2 \cdot 3^3\). Here, \(3^3\) means 3 is raised to the power of 3, which is an exponentiation operation. This means you need to multiply 3 by itself three times.

Step 2: Evaluate the Exponent in Part a

Calculate \(3^3\), which means multiplying 3 by itself two more times: \(3 \cdot 3 \cdot 3 = 9 \cdot 3 = 27\).

Step 3: Perform the Multiplication in Part a

Now that we have evaluated the power, multiply the result by 2: \(2 \cdot 27 = 54\).

Step 4: Understanding the Expression for Part b

The expression given for part b is \((2 \cdot 3)^3\). Notice the parentheses in the expression. This indicates that you need to perform the operation inside the parentheses first, according to the order of operations (PEMDAS/BODMAS).

Step 5: Calculate the Expression Inside Parentheses for Part b

Inside the parentheses, multiply 2 by 3: \(2 \cdot 3 = 6\).

Step 6: Evaluate the Exponent in Part b

Now, raise the result to the power of 3: \(6^3\). This means \(6 \cdot 6 \cdot 6 = 36 \cdot 6 = 216\).

Key Concepts

Order of Operations

When solving mathematical expressions, it's crucial to follow the order of operations. This order dictates the sequence in which different operations should be performed to ensure accurate results. Commonly remembered by the acronym PEMDAS (or BODMAS in some countries), this stands for:

• Parentheses (Brackets)
• Exponents (Orders)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

When you encounter a complex expression, always start with calculations inside the parentheses, then address any exponents.
Proceed with multiplication or division, and finally, execute addition or subtraction.
This structured method helps avoid incorrect solutions and provides a clear pathway through mathematical problems.

Multiplication

Multiplication is one of the core operations in mathematics. It involves combining equal groups of objects or numbers to find out how many there are in total. For instance, when you see "2 \( \cdot \) 3," this indicates two groups of three objects.
Multiplication is often faster and more efficient than repeated addition, especially when dealing with larger numbers. For example, multiplying 27 by 2 directly gives you 54, which was calculated from the operation in Part a of the original exercise.
Multiplication here followed the calculation of an exponent. Knowing about multiplication helps in smoothly transitioning between using different operations, particularly after evaluating exponents or numbers inside parentheses.

Parentheses in Mathematics

Parentheses play a crucial role in organizing and simplifying mathematical expressions. They are used to group parts of an expression that should be handled first before applying other operations. In expressions like \( (2 \cdot 3)^3 \), the parentheses tell us to focus on the operation inside first: Multiply 2 by 3 before raising the result to a power.
In absence of parentheses, the order of operations (PEMDAS/BODMAS) would dictate the sequence, but parentheses always take precedence, overriding the usual order.
Utilizing parentheses effectively allows for clear expression management and aids in avoiding errors, ensuring calculations are performed in the intended logic sequence. This is vital for tackling more complex math problems efficiently.