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Chapter 0: Problem 2

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

Short Answer

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

Step by step solution

01

Evaluate the exponent

Evaluate \(6^{2}\) first according to the order of operations, which gives \(6^{2} = 36\)
02

Multiply the result by 2

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section.Exercises \(144-146\) will help you prepare for the material covered in the next section. Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator. $$ \frac{x^{2}+6 x+5}{x^{2}-25} $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) toproduce a true statement. $$\left(4 \times 10^{3}\right)+\left(3 \times 10^{2}\right)=4.3 \times 10^{3}$$

Why is \(\left(-3 x^{2}\right)\left(2 x^{-5}\right)\) not simplified? What must be done to simplify the expression?

Simplify by reducing the index of the radical. $$\sqrt[9]{x^{6}}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using my calculator, I determined that \(6^{7}=279,936,\) so 6 must be a seventh root of \(279,936\).
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