Problem 90 - Algebra for College Students

Question

How would you simplify \(\left(2^{3}+2^{2}\right)^{2} ?\) Explain your reasoning.

Step-by-Step Solution

Answer

The simplified expression is 144.

Step 1: Evaluate Exponents Inside the Parentheses

First, we need to evaluate the exponents inside the parentheses: calculate \(2^3\) and \(2^2\).\[2^3 = 8\]\[2^2 = 4\]

Step 2: Simplify Inside the Parentheses

Add the results from Step 1 together.\[8 + 4 = 12\]

Step 3: Apply the Square to the Result

Now that we have simplified the expression inside the parentheses to \(12\), we need to square it.\[12^2 = 144\]

Step 4: Final Result

The simplified form of the expression is 144.

Key Concepts

Evaluating Exponents

Understanding exponents is essential when simplifying algebraic expressions. An exponent tells us how many times to multiply a number, called the base, by itself. For example, in the expression \(2^3\), 2 is the base, and 3 is the exponent. This means we multiply 2 by itself three times, which is \( 2 \times 2 \times 2 = 8 \). Similarly, \(2^2\) means we multiply 2 by itself once more, resulting in \(2 \times 2 = 4\). Evaluating exponents accurately is crucial to ensure calculations proceed smoothly. When you're dealing with multiple exponents within a parenthesis, always evaluate each one separately before proceeding with any additional algebraic operations. This foundational step is crucial for accurately simplifying expressions, especially when these exponents play a role further down in the calculations.

Order of Operations

The order of operations is like a roadmap guiding us on the correct path to solve mathematical problems accurately. It ensures consistency across different people and situations when simplifying expressions. The widely recognized rule is PEMDAS, which stands for:

• P: Parentheses first
• E: Exponents (powers and roots, etc.)
• M/D: Multiplication and Division (left-to-right)
• A/S: Addition and Subtraction (left-to-right)

In the given expression \( \left(2^3 + 2^2\right)^2 \), we apply this rule by first evaluating the expressions within the parentheses. After evaluating the exponents, proceed with addition or subtraction within the parentheses. Finally, consider the exponent outside the parentheses, which in this case, means squaring the result. This sequenced approach aids in avoiding common mistakes and achieving the correct result.

Basic Math Simplification

Simplification in math means reducing an expression to its simplest form. This involves carrying out all possible mathematical operations within the constraints of the order of operations. In the given task, simplification began by evaluating exponents within the parentheses. The intermediate result \(2^3 = 8\) and \(2^2 = 4\) was then added to give 12. Once the parentheses were simplified, the last step was to square this outcome, obtaining \(12^2 = 144\). It's important to ensure each part is completed one step at a time without jumping ahead. Always remember:- After evaluating exponents, always proceed to simplify the results within parentheses.- Follow the structured order: work inside out from any nested layers, just like peeling an onion.This methodical approach helps in handling complex expressions with ease, leading to correct results faster with fewer errors.