Problem 63 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$-2^{2}+4[16 \div(3-5)]$$

Step-by-Step Solution

Answer

The simplified form of the expression -2^{2}+4[16÷(3-5)] is -36.

Step 1: SOLVE INSIDE THE BRACKETS

-2^{2}+4[16 ÷(3-5)] simplifies to -2^{2}+4[16÷(-2)]. We subtract three from five to get -2.

Step 2: PERFORM THE DIVISION IN THE BRACKETS

Then the resulting expression -2^{2}+4[16÷-2] becomes -2^{2}+4[-8] by dividing 16 by -2.

Step 3: MULTIPLY INSIDE THE BRACKETS

The resulting expression -2^{2}+4[-8] becomes -2^{2} - 32 after multiplying 4 by -8.

Step 4: CALCULATE THE EXPONENTIAL OPERATOR

Then, the expression -2^{2} - 32 turns into -4 - 32, as the square of -2 is 4, but due to the preceding negative sign, it becomes -4.

Step 5: PERFORM ADDITION AND SUBTRACTION FROM LEFT TO RIGHT

Finally, the expression -4 - 32 is equal to -36 by subtracting 32 from -4.

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value. This process makes mathematical expressions easier to work with and understand. A simplified expression is free of complex fractions, unnecessary parentheses, and is as concise as possible.

When simplifying, we often follow the order of operations, which is crucial to getting the correct result. The order of operations provides a consistent framework for solving expressions, helping us know which operations to complete first. Let's remember our helpful mnemonic: PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

• Always resolve operations within parentheses first.
• Next, handle any exponents in the expression.
• Then, perform all multiplication and division.
• Finally, carry out addition and subtraction.

By following this procedure, your expressions will be simplified accurately.

Integer Operations

Integer operations involve performing basic arithmetic like addition, subtraction, multiplication, and division with whole numbers (integers). These fundamental skills are essential in algebra.

When dealing with integer operations, keep in mind:

• Adding a positive number increases the value.
• Adding a negative number decreases the value.
• Subtracting an integer is the same as adding its opposite.
• Multiplying or dividing two integers with the same sign results in a positive integer.
• Multiplying or dividing two integers with different signs results in a negative integer.

For example, if we have the expression \(16 \div -2\), we recognize this as a division operation where the dividend (16) is divided by the divisor (-2). Since both numbers have different signs, the result is negative, giving us \(-8\). Understanding these operations is crucial for accurately simplifying expressions.

Exponents

Exponents are a way of expressing repeated multiplication of the same number. For instance, \(2^3\) means \(2 \times 2 \times 2\), which equals \(8\).

In our example, we have \(-2^2\). An important distinction here is how the negative sign is treated. The exponent only affects the number it is directly attached to unless parentheses specify otherwise.

• Without parentheses, \(-2^2\) is the same as \(-(2^2)\), equating to \(-4\).
• With parentheses, \((-2)^2\), it would result in \(4\) because the entire term, including the negative sign, is squared.

Understanding how to correctly interpret and calculate expressions with exponents is an essential skill in algebra basics, as it influences the results significantly.

Algebra Basics

Algebra basics form the foundation of more advanced mathematical understanding, involving symbols and letters that represent numbers in expressions and equations. The goal is to manipulate these symbols to solve problems, find unknown values, or simplify expressions.

Key aspects of algebra basics include:

• Understanding variables and constants.
• Applying arithmetic operations to algebraic expressions.
• Utilizing the order of operations to simplify expressions.
• Recognizing patterns and structures within expressions to solve equations efficiently.

In our exercise, we applied algebra basics to ensure we simplified the expression correctly while respecting all mathematical rules. With enough practice, mastering these fundamentals will make solving complex problems easier and more intuitive.