Problem 64 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$-3^{2}+2[20 \div(7-11)]$$

Step-by-Step Solution

Answer

-1

Step 1: Calculate Squares and Solve the parentheses

First, calculate the square of -3, which is \((-3)^{2} = 9 \). Next, solve the operations within the parentheses, that is \(7-11 = -4\)

Step 2: Simplify Expressions in the Brackets

Then, simplify the contents within the brackets by using the division operation \(20 \div -4\) = \(-5\). Then remultiply this by 2 to get the second term as \(2 \times (-5) = -10\). Remember that the multiplication operation is also included in the brackets operations.

Step 3: Perform the Final Addition Operation

Finally, perform the addition operation, which is \(9 + (-10) = -1\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. They form the basis of algebra and help represent real-world situations in a mathematical format. When dealing with algebraic expressions, it is important to understand that variables can represent unknown values, making it possible to express and solve problems more abstractly.

In the expression provided, \(-3^{2} + 2[20 \div (7-11)]\), numbers and operations are combined strategically to form a meaningful expression. Here's a brief breakdown of key components:

• \(-3^{2}\): The square of \(-3\), which tells us to multiply \(-3\) by itself.
• \(2[20 \div (7-11)]\): A more complex part involving brackets and divisions, signaling that different operations need to be followed in a specific order.

Algebraic expressions such as this one allow flexibility and tremendous power in mathematics as they can be simplified, evaluated, or solved under various conditions.

Simplifying Expressions

Simplifying expressions means performing all possible mathematical operations to reduce an expression to its simplest form. This process involves following the order of operations, also known as PEMDAS/BODMAS rules (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order ensures consistent results in mathematical calculations.

In the original exercise, simplifying the expression \(-3^{2} + 2[20 \div (7-11)]\), involves performing each operation step-by-step:

1. **Calculate the exponent:** - \(-3^{2}\) equals 9. In this case, since it's inside the square and not outside, the negative sign is not included in the squaring process.2. **Solve the parenthesis:** - Inside the parentheses, calculate \(7 - 11\) resulting in \(-4\).3. **Perform division within brackets:** - \(20 \div -4 = -5\).4. **Follow multiplication inside brackets:** - Multiply 2 by \(-5\) to get \(-10\).5. **Finish with addition/subtraction:** - Finally add 9 to \(-10\) which results in \(-1\).

This systematic approach guarantees the expression is reduced correctly and elegantly.

Mathematical Operations

Mathematical operations are the actions taken to solve expressions or equations, such as addition, subtraction, multiplication, division, and exponentiation. Understanding how these operations interact within algebraic expressions is crucial for solving them accurately.

The provided expression involves several operations, each needing to be tackled in a specific sequence:

• **Exponentiation:** Calculate powers first, e.g., solving \(-3^{2}\).
• **Parentheses/Brackets:** Solve what's inside these first, for e.g., \(7 - 11 = -4\).
• **Division and Multiplication:** Done from left to right. Here, \(20 \div -4 = -5\) followed by multiplying by 2.
• **Addition or Subtraction:** Finally performed after other operations, such as adding \(9\) and \(-10\).

These operations are organized carefully to yield the correct result, in this case, a final value of \(-1\). Remembering which operation to apply and when is essential, as it affects the outcome of any mathematical task.