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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

For each expression, use parentheses to group the numbers together and to group the powers with like bases together. Example: \(a \cdot 4 \cdot a^{3} \cdot 2=(4 \cdot 2)\left(a \cdot a^{3}\right)\) $$12 \cdot 15 \cdot a \cdot 9 \cdot a^{5} \cdot c^{3}$$

For each expression, use parentheses to group the numbers together and to group the powers with like bases together EXAMPLE \(: a \cdot 4 \cdot a^{3} \cdot 2=(4 \cdot 2)\left(a \cdot a^{3}\right)\) \(b \cdot 5 \cdot 10 \cdot b^{4}\)

Find each product or quotient. Express using exponents. $$x^{2} \cdot x^{4}$$

Evaluate each expression if \(x=10, y=-5,\) and \(z=4 .\) Write as a fraction simplest form. $$\frac{1}{z y}$$

Write and solve an equation to find each number. The sum of a number and 9 is \(-2\)
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