Problem 2 - College Algebra

Question

Evaluate each exponential expression. $$ 6^{2} \cdot 2 $$

Step-by-Step Solution

Answer

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

Step 1 - Evaluate the Exponential Expression

Begin by calculating the exponential part of the expression \(6^{2}\), which means 6 multiplied by itself. It results in 36.

Step 2 - Multiply the Result by 2

After the exponent has been calculated, continue by multiplying the result from step 1 with 2 according to the order of operations. So, \(36 \cdot 2 = 72\).

Key Concepts

Understanding Order of Operations

When dealing with mathematical expressions, we must follow a specific set of rules known as the order of operations to ensure that everyone arrives at the same answer. These rules help us know which part of an expression to solve first. In simpler terms, they tell us the sequence in which different mathematical operations should be carried out. In our exercise, we see both an exponent and a multiplication. It’s crucial to know that exponents should be evaluated before multiplication.

The basic order of operations is often remembered with the acronym PEMDAS:

• P for Parentheses
• E for Exponents
• M/D for Multiplication and Division (from left to right)
• A/S for Addition and Subtraction (from left to right)

By following this order, you will solve the expression correctly. It ensures that you handle the exponential part before multiplying with other numbers, which was crucial in evaluating \(6^2 \cdot 2\).

Evaluating Exponents

Before we can move on to other operations, let's understand exponents. An exponent is a shorthand way to show repeated multiplication. It consists of a base number and an exponent, which tells us how many times to use the base as a factor. For example, \(6^2\) means that 6 should be multiplied by itself. So, \(6^2\) equals \(6 \times 6\), and this gives us 36.

When you encounter an exponential expression within a longer expression, like \(6^2 \cdot 2\), you must calculate the exponential expression first before proceeding. Successfully evaluating the exponent ensures that you can apply the order of operations accurately, solving simpler portions first and more complex operations last. This method helps in avoiding small mistakes that could change the outcome significantly.

Multiplying Numbers After Exponents

Once you have handled the exponents, the next step is to perform multiplication. In our exercise, after evaluating \(6^2\) as 36, you then multiply 36 by 2 to continue solving the expression. Multiplication is a basic arithmetic operation where a number is added to itself a certain number of times.

For \(36 \cdot 2\), think of it as adding 36 together two times: \(36 + 36\), which equals 72. Multiplying after properly evaluating exponents ensures that you adhere to the order of operations, arriving at the correct answer. This process might seem simple, but each step reinforces a fundamental understanding of how these operations interact within mathematical expressions.