Problem 51 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement using \(<\) or \(>\). \((6 \cdot 3)^{3} ? 6 \cdot 3 \cdot 3\)

Step-by-Step Solution

Answer

Therefore, the complete statement is \((6 \cdot 3)^{3} > 6 \cdot 3 \cdot 3\).

Step 1: Calculate the Left Side

First, calculate the expression on the left side of the question mark. For this expression \((6 \cdot 3)^{3}\), solve the calculation inside the parentheses first - which is \(6 \cdot 3 = 18\), then raise this result to the power of 3: \(18^{3} = 5832\). So, the result of the left side is 5832.

Step 2: Calculate the Right Side

Now, calculate the right side of the question mark. This is a simple multiplication operation: \(6 \cdot 3 \cdot 3\). Multiply all of these numbers together to get 54.

Step 3: Compare the Results

Now with both sides calculated, it's time to fill in the question mark with the correct comparative operator. Because 5832 is greater than 54, the correct comparative operator to use is \(>\).

Key Concepts

Order of Operations

When solving mathematical expressions, it is crucial to follow the order of operations to ensure accurate results. This hierarchy dictates the sequence in which different parts of an equation should be solved. According to this convention, the acronym PEMDAS is commonly used to remember the sequence: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

To apply this correctly, look at the original problem: Copy and complete the statement using < or >. \( (6 \cdot 3)^{3} ? 6 \cdot 3 \cdot 3\). Following the order of operations, we start with the parentheses. This means we calculate \(6 \cdot 3\) first. If we had other operations, such as addition or subtraction, those would have to wait until after any parentheses or exponents are taken care of. Only then can we progress to multiplication, division, and finally addition or subtraction if they are present in the expression.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number that is multiplied by itself, and the exponent denotes how many times the base is used as a factor in the multiplication.

In the problem we are considering, \( (6 \cdot 3)^{3} \), \(6 \cdot 3\) is calculated first to get 18. Then, exponentiation is applied to the result, which means 18 is used as the base and is multiplied by itself two more times (\(18 \cdot 18 \cdot 18\)), as indicated by the exponent 3. This result is significantly different from simply multiplying the original numbers without the exponent, illustrating how powerful exponentiation is when compared to standard multiplication.

Inequalities

Inequalities are used to compare the sizes of two expressions and establish relationships such as less than, greater than, or equal to. Comparing two sides of an equation using inequalities allows us to understand how two sets of numbers relate to each other without finding precise values.

In the context of the given problem, after computing both expressions, we obtain two results: \( 5832 \) and \( 54 \). We then use the appropriate inequality symbol to compare these two numbers: 5832 is greater than 54, which is denoted by \(>\). Understanding inequalities is essential for solving and graphing equations, and for conceptualizing the range of possible solutions in real-world situations.