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Chapter 4: Problem 9

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

Short Answer

Expert verified
The expression evaluates to 50.

Step by step solution

01

Calculate the Power

First, calculate the value of the power in the expression. The expression given is \( 2 \cdot 5^{2} \). This means you need to square the number 5 first.\[ 5^{2} = 5 \times 5 = 25 \]
02

Multiply the Base

Now, use the result from Step 1 and multiply it with the coefficient (the number 2) in front of the power.\[ 2 \cdot 25 = 50 \]

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Key Concepts

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

Order of Operations
When working with mathematical expressions, it's important to follow the order of operations to solve problems accurately. This order is commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This set of rules ensures that everyone evaluates expressions the same way, providing consistency across mathematical problems.

Let's break this down to make it simpler:
  • Parentheses: First, solve expressions within parentheses or brackets.
  • Exponents: Next, calculate powers like squares or cubes.
  • Multiplication and Division: Then move from left to right performing all multiplication and division.
  • Addition and Subtraction: Lastly, go from left to right adding or subtracting numbers.
In the exercise given, there are no parentheses, so the first step is to evaluate the exponents. After that, you proceed with multiplication. The order of operations helps ensure we get the correct final result.
Exponents
Exponents are a way to represent repeated multiplication of the same number by itself. In mathematical notation, an exponent is displayed as a small number raised to the upper right of a base number. For instance, in the expression \(5^2\), the number 5 is the base, and 2 is the exponent.

This tells us to multiply 5 by itself once, which is \(5 \times 5 = 25\). Exponents make it easier to write and work with large numbers and expressions by simplifying lengthy multiplication processes.
  • Understanding: Think of exponents as a shorthand way to describe multiplication repeated a certain number of times.
  • Use in Expressions: In our expression \(2 \cdot 5^2\), the exponent needs to be calculated before you proceed to any further operations.
Mastering exponents is crucial as they appear in many areas of mathematics, including algebra, calculus, and more.
Multiplication
Multiplication is one of the four fundamental arithmetic operations and extends the idea of repeated addition. It involves finding the total of one number (the multiplicand) taken a specified number of times (the multiplier). In our expression, once we have calculated the exponent's value, we need to multiply.

Here鈥檚 a quick breakdown of the process:
  • Using Results: From our exercise, we determined that \(5^2 = 25\).
  • Perform Multiplication: Then, multiplying the obtained result with 2 gets \(2 \times 25 = 50\).
Multiplication simplifies and speeds up the addition of numbers, making it easier to handle large quantities. It is frequently used in everyday calculations and forms the foundation for more advanced mathematical concepts. Understanding multiplication and when to use it is a key skill in solving various math problems.

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Most popular questions from this chapter

Which One Doesn't Belong? Identify the fraction that does not belong with the other three. Explain your reasoning. $$\frac{6 y}{5}$$ $$\frac{4}{7}$$ $$\frac{1}{x^{2}}$$ $$\frac{15}{12}$$

Use the Distributive Property to write each expression as an equivalent expression. $$(b-3)(-2)$$

Factor each monomial. $$-70 x y z$$

Find each product. $$4.05 \times 10$$

Simplify each fraction. If the fraction is already in simplest form, write simplified. $$\frac{40 d}{42 d}$$
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