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Chapter 0: Problem 1

Evaluate each exponential expression. $$5^{2} \cdot 2$$

Short Answer

Expert verified
The evaluated value of the expression \(5^{2} \cdot 2\) is 50.

Step by step solution

01

Evaluate the Exponential Expression

Start by computing the value of the exponential expression \(5^{2}\). According to the exponent rules, \(5^{2}\) means \(5 \times 5\). As a result, \(5^{2} = 25\). The original expression now becomes \(25 \cdot 2\).
02

Perform Multiplication

Next, proceed to perform the multiplication. Multiply 25 by 2, which results in 50. So, \(25 \cdot 2 = 50\).
03

Final Answer

Hence, the value of the exponential expression \(5^{2} \cdot 2\) is 50.

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Key Concepts

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

Exponent Rules
When dealing with exponential expressions, it's crucial to understand the core concepts of exponent rules. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, in the expression \(5^2\), the base is 5, and the exponent is 2. This tells us to multiply 5 by itself.
So, \(5^2\) translates to \(5 \times 5\), which equals \(25\).
  • The base: The number that you multiply by itself.
  • The exponent: The small number written directly above and to the right of the base; it tells you how many times to multiply the base by itself.
Understanding this principle helps when simplifying expressions or solving equations that involve powers.
Multiplication
After simplifying the exponential part of the expression, multiplication is often the next step. In our example, we took \(5^2\) and simplified it to 25. Now the expression looks like \(25 \cdot 2\). The dot between the numbers typically represents multiplication.
Multiplication is one of the basic arithmetic operations, which involves calculating the total of one number repeated a specified number of times, defined by the other number. In this case, you multiply 25 by 2. Hence, the operation is performed as follows:
  • Multiply the numbers directly: \(25 \times 2 = 50\).
  • Check your work to ensure you perform the correct multiplication sequence.
This straightforward process helps us transition from the exponential expression to a final numerical result.
Step-by-Step Solution
Breaking down problems into step-by-step solutions is essential for understanding complex mathematical concepts.
In our given problem, we evaluated \(5^2 \cdot 2\) by first applying the exponent rule, which involved calculating \(5^2\). We derived this to equal 25.
Next, we took this result and applied the multiplication operation by multiplying \(25 \cdot 2\), which resulted in 50.
  • Step 1: Calculate the power \(5^2 = 25\).
  • Step 2: Multiply the result by 2, so \(25 \times 2 = 50\).
  • Step 3: Conclude by stating the final result of the expression \(5^2 \cdot 2 = 50\).
Each step reinforces the understanding of different mathematical operations and how they integrate to solve a problem.

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Most popular questions from this chapter

Describe how to solve an absolute value inequality involving the symbol \(>.\) Give an example.

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{5}{4} \cdot \frac{8}{15}$$

Solve each equation. $$\frac{1}{x^{2}-3 x+2}=\frac{1}{x+2}+\frac{5}{x^{2}-4}$$

Solve each equation. $$\begin{aligned} 45-[4-2 x-4(x+7)] &=\\\ -4(1+3 x)-[4-3(x+2)-2(2 x-5)] \end{aligned}$$

Will help you prepare for the material covered in the next section. Multiply and simplify: \(12\left(\frac{x+2}{4}-\frac{x-1}{3}\right)\)
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