Problem 2 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$ 6^{2} \cdot 2 $$

Step-by-Step Solution

Answer

The value of the expression \(6^{2} \cdot 2\) is 72.

Step 1: Evaluate the exponent

Evaluate \(6^{2}\) first according to the order of operations, which gives \(6^{2} = 36\)

Step 2: Multiply the result by 2

Then, multiply the result from the first step by 2 which gives \(36 * 2 = 72\)

Key Concepts

Order of Operations

When faced with an expression like \(6^{2} \cdot 2\), it's crucial to know in what sequence to tackle the elements. This is where the 'order of operations' plays a crucial role. The order of operations acts like a set of instructions that tells you what part of a mathematical problem to solve first.

The most widely accepted order of operations is remembered by the acronym PEMDAS:

• Parentheses
• Exponents (i.e., powers and roots, such as squares and square roots)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

By adhering to these rules, you ensure that your solution is consistent with correct mathematical practices. Thus, according to PEMDAS, you would handle the exponent before addressing the multiplication.

Evaluate Exponents

An exponent tells you how many times to multiply a number by itself. In our problem, \(6^2\) means you multiply 6 by itself two times: \(6 \times 6\).

This process is called 'evaluating the exponent,' and it's a key part of tackling any equation containing exponential expressions. By evaluating the exponent first, you transform a more complicated expression into something simpler and easier to handle. For instance:

• 6 multiplied by itself: \(6 \times 6 = 36\)

This simplification makes the subsequent steps more straightforward. By breaking it down this way, exponents become less of an intimidating mystery and more like standard multiplication.

Multiplication

Multiplication is the process of taking one number and adding it to itself a certain number of times, determined by another number. After evaluating an exponent, the next step in the order of operations often involves multiplying the result.

In our example, after evaluating \(6^2\) to get 36, we then move to the multiplication step. This involves multiplying 36 by 2:

• The expression \(36 \times 2\) becomes simple multiplication.
• We add 36 to itself two times or, simply, calculate \(36 \times 2 = 72\).

Multiplication turns repeated addition into a more efficient process. By understanding multiplication in this way, you're better equipped to handle exponential expressions that require this operation as the final step.