Problem 12 - College Algebra

Question

Simplify the given expression. $$ (4+5)^{2} \div 3 $$

Step-by-Step Solution

Answer

The simplified result is 27.

Step 1: Solve the Parentheses

First, simplify the expression inside the parentheses: \(4 + 5 = 9\). So, the expression becomes \((9)^2 \div 3\).

Step 2: Apply the Exponent

Next, solve the exponent by squaring 9: \(9^2 = 81\). This makes the expression \(81 \div 3\).

Step 3: Divide the Result

Finally, divide 81 by 3: \(81 \div 3 = 27\).

Key Concepts

Simplification

Simplification is the process of expressing a mathematical expression in its most straightforward form. This involves following a set of rules for operations to ensure accuracy in the final result. By simplifying expressions, we reduce complexity for easier reading and solving.

• It's key to break down expressions step by step.
• Follow the standard order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Always start with the innermost operation and move outward.

The goal of simplification is to make complex problems more manageable while ensuring nothing is lost in translation.

Parentheses

Parentheses play a crucial role in determining the order in which calculations should be performed. They are used to group parts of a mathematical expression, indicating that the operations within are to be completed first.

• Consider what's inside the parentheses as a priority, similar to tackling the task that comes first.
• By resolving the parentheses, you make the rest of the expression easier to handle.

In the expression \[(4+5)^2 \div 3\], the operation inside the parentheses is 4 plus 5, which simplifies to 9. This simplification is the first step toward solving the entire expression efficiently.

Exponents

Exponents are a shorthand way to express repeated multiplication of a number by itself. When dealing with exponents, you always solve these after the parentheses and before other operations, following the order of operations.

• An exponent indicates how many times to use the number in a multiplication.
• For example, in \(9^2\), the number 9 is multiplied by itself, yielding 81 as a result.

Including exponents in your calculations can significantly change the quantity you end up with since even a small base number can become large rapidly with higher powers.

Division

Division is the process of distributing a number into equal parts or groups, and it simplifies expressions by reducing them into a single value. In mathematics, division follows directly after exponents in the order of operations.

• It is performed after any necessary parentheses and exponent calculations have been solved.
• In our expression \(81 \div 3\), division is the final step, reducing everything to a simple number 27.

Remember, while division sounds straightforward, it significantly impacts the outcome if not executed correctly. It's important to ensure all preceding operations are completed before moving on to division.