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Chapter 1: Problem 81

Evaluate. $$5^{2} \cdot 2 \cdot 3^{4}$$

Short Answer

Expert verified
The result of evaluating \(5^{2} \cdot 2 \cdot 3^{4}\) is 4050.

Step by step solution

01

Compute The Exponents

Evaluate the expressions with exponents before proceeding to multiplication. This means \(5^{2}\) and \(3^{4}\) should be calculated first. Using the rule of exponents, \(5^{2} = 25\) and \(3^{4} = 81\).
02

Perform The Multiplication

After calculating all the exponents, proceed to multiply all these results together. This means multiplying 25 (from \(5^{2}\)), 2, and 81 (from \(3^{4}\)). This gives \(25 \times 2 \times 81 = 4050\).
03

Final Result

After multiplying all the results, we find that \(5^{2} \cdot 2 \cdot 3^{4} = 4050\).

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Key Concepts

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

Understanding Multiplication in Expressions
Multiplication is one of the basic arithmetic operations used in simplifying mathematical expressions.
When faced with a problem like the one in our exercise, multiplication is applied to combine results from calculations.

Let's break it down simply:
  • Multiplication involves finding the total number of objects in a certain number of groups, similar to repeated addition.
  • In the expression given, multiplication connects the values we get from evaluating each exponent.
  • In our exercise, this involves multiplying 25, 2, and 81.
Imagine putting these numbers together: You first multiply 25 by 2, resulting in 50.
Then multiply that 50 by 81 to yield 4050.
This process unites parts of an expression into a cohesive result.
Order of Operations: PEMDAS Explained
The order of operations is crucial in mathematics to ensure consistency and accuracy.
PEMDAS is the acronym that helps us remember this order: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

In our exercise, the expression was evaluated using these rules:
  • First, calculate the exponents (\(5^{2}\) and \(3^{4}\)) before any multiplication.
  • Ignoring this order can lead to incorrect results, as interpreting mathematical expressions can differ without standard rules.
This clarity benefits everyone working with math, from professionals to students.
It removes ambiguity and results in the expected outcome, just like our correctly simplified solution of 4050.
Evaluating Expressions Step by Step
Evaluation in math refers to finding the value of an expression by performing necessary calculations.
It's a step-by-step process, and it's crucial to follow logical sequencing, as demonstrated in our exercise.

Here's how you can approach evaluation:
  • Step 1: Identify and tackle any exponents. We calculated \(5^{2} = 25\) and \(3^{4} = 81\) first.
  • Step 2: Focus then on the multiplication, combining values in the correct sequence. This resulted in \(25 \times 2 = 50\), moving next to \(50 \times 81 = 4050\).
  • Step 3: Reveal the solution. The end value of 4050 confirms you've simplified and evaluated all parts accurately.
Always go through each step without rushing.
This careful evaluation ensures an accurate answer, reinforcing your understanding of mathematics.

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