Problem 73 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ (-3-\sqrt{25})^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 64.

Step 1: Simplify Inside the Parentheses

First, simplify the expression inside the parentheses, \(-3 - \sqrt{25}\). Since \(\sqrt{25} = 5\), the expression becomes \(-3 - 5\).

Step 2: Calculate the Sum Inside the Parentheses

Compute the sum from the previous step: \(-3 - 5 = -8\).

Step 3: Square the Result

Now take the result from Step 2 and square it: \((-8)^{2}\).

Step 4: Final Calculation

Calculate \((-8)^{2}\). This is \(-8 \times -8\), which equals \(64\).

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making them easier to work with. When you have an expression, your goal is to reduce it to its simplest form. This means performing operations and combining like terms.

For the given exercise, start by focusing on the part inside the parentheses, which is essential for simplification. Look at \(-3 - \sqrt{25}\). **Why?** Because simplifying inside the parentheses allows you to work with a much simpler expression. After determining that \(\sqrt{25} = 5\), replace it, turning it into \(-3 - 5\).

Finally, simplify \(-3 - 5\) to \(-8\). This step is crucial to make further calculations easier. Simplifying expressions lays the foundation for correctly evaluating them, and it ensures accuracy in complex mathematical problems.

Squares and Roots

In mathematics, squares and roots are inverse operations. They help in simplifying and solving equations. A square of a number is the number multiplied by itself. For example, if you have the expression \((-8)^2\), you multiply \(-8\) by itself to get \(64\).

Square roots are operations that "undo" squaring a number. They ask us, which number multiplied by itself gives us the original number? For instance, the square root of 25 is 5, because \(5 \times 5 = 25\). In our exercise, understanding \(\sqrt{25}\) was necessary to simplify \(-3 - \sqrt{25}\) into \(-3 - 5\).

By grasping the relationship between squares and roots, you can perform operations more effectively. Recognizing these relationships often speeds up your problem-solving process and ensures open doors to more advanced topics in mathematics.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. These are the basic building blocks of mathematics, helping to solve problems and simplify expressions. In our original exercise, you encounter subtraction and multiplication.

Start by subtracting carefully inside the parentheses. Take \(-3\) and subtract 5 to get \(-8\). Each of these arithmetic operations historically links to a real-world situation, making them intuitive once you follow the rules accurately.

Then multiply when squaring \(-8\). Remember, squaring \((-8)^2\) is similar to doing \(-8 \times -8\). **Important thing to note is sign behavior:** negative times negative equals positive, thus getting 64. Arithmetic operations form the core tools you'll need to tackle any expression or equation, paving the way for deeper mathematical exploration.