Problem 30 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The simplified expression is 50.

Step 1: Evaluate the Exponent

The first operation is to evaluate the exponent. We have \(5^2\), which means \(5\) multiplied by itself. This gives us \(5 \times 5 = 25\).

Step 2: Perform Multiplication

Once the exponent is evaluated, we need to perform the multiplication with the remaining part of the expression. Multiply \(2\) by \(25\) from Step 1. So, \(2 \times 25 = 50\).

Step 3: Conclusion

The simplified form of the expression \(2 \cdot 5^{2}\) is \(50\).

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making a math problem easier to work with by reducing or "condensing" it into a simpler form. In algebra and arithmetic, this often means taking something complex, with operations like exponents or multiplication, and breaking it down step-by-step.When you see an expression like \(2 \cdot 5^{2}\), your goal is to get this down to a single number if possible. Simplifying involves:

• Identifying what operations are involved—like addition, subtraction, multiplication, and division—and handling them in the right order.
• Calculating the values of any powers or exponents first before moving on to the next operation.
• Eventually performing any remaining operations, like multiplication or addition.

By simplifying expressions, you make it easier to understand the problem and solve it quickly. It also helps to verify the result by making sure everything adds up correctly, and nothing gets left out or miscalculated.

Order of Operations

The order of operations is critical when simplifying expressions. This order ensures that you perform calculations in the correct sequence to arrive at the right answer.A common mnemonic for remembering the order is PEMDAS:

• Parentheses first: Solve expressions inside parentheses.
• Exponents next: Evaluate powers and roots.
• Multiplication and Division: Proceed from left to right.
• Addition and Subtraction: Also from left to right.

In the expression \(2 \cdot 5^2\), we don’t have parentheses, so we start by evaluating the exponent \(5^2\) first, as dictated by the rule "Exponents come before Multiplication." This gives us \(25\), and then we multiply by \(2\) to get our final result of \(50\). Remember, mixing up the order can lead to a completely different outcome, so it's important to follow these operations strictly.

Multiplication Concepts

Understanding multiplication is crucial in handling expressions like \(2 \cdot 5^2\). Multiplication is essentially repeated addition—adding a number (the multiplicand) to itself a certain number of times indicated by the multiplier.In our expression, after evaluating the exponent to get \(25\), you're multiplying this result by \(2\). Here's how multiplication helps:

• Efficiency: Instead of having to add \(25\) to itself two times separately, multiplication lets you do it in one step.
• Clarity: When you use multiplication, the operation is clearer and often more concise.

The simplified multiplication here shows how we take those steps effortlessly and accurately by recognizing that multiplication compresses repeated addition into a single, easy operation. It’s a fundamental math skill that expedites calculations and simplifies complex problems into manageable tasks.