Question

For Exercises \(1-74,\) simplify. $$(5-7)^{2} \div(-2)-(-3) \div 3$$

Short answer

Question: Simplify the expression \((5 - 7)^{2} \div(-2) - (-3) \div 3\). Answer: -1

Step-by-step solution

Apply Exponents

We need to deal with the exponent first. In this case, find the value of \((5 - 7)^{2}\). The expression inside the parentheses is \((5 - 7)\), which equals \(-2\). Then, raise this to the power of 2: \((-2)^{2}\).$$(-2)^{2} = (-2 \times -2) = 4$$Now, the expression becomes:$$4 \div(-2) - (-3) \div 3$$

Perform Division Operations

Next, we need to perform the division operations. In this expression, there are two divisions: \(4 \div(-2)\) and \(-3 \div 3\). Divide the numbers as follows:$$4 \div (-2) = -2$$$$-3 \div 3 = -1$$Now, the expression becomes:$$-2 - (-1)$$

Simplify the Final Expression

The final expression is \((-2) - (-1)\). Remember that subtracting a negative number is equivalent to adding its positive counterpart. Therefore, we can rewrite the expression as:$$-2 + 1$$Now, perform the addition:$$-2 + 1 = -1$$The simplified expression is \(-1\).

Key concepts

Order of Operations

When simplifying algebraic expressions, it's crucial to follow the order of operations, a set of rules known as PEMDAS or BIDMAS. This acronym stands for Parentheses (or Brackets), Exponents (or Indices), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Here's how the order of operations applies to a problem like the exercise above:

• Parentheses (Brackets) – Evaluate expressions inside parentheses first. In the exercise, we start with \( (5-7)^2 \).
• Exponents (Indices) – Next, we address any exponents. This is where \( (-2)^2 \) is calculated, based on the result of the parentheses.
• Multiplication/Division – Then, we perform any multiplication or division from left to right. For the exercise, it's \( 4 \div (-2) \) and \( (-3) \div 3 \).
• Addition/Subtraction – Finally, solve any addition or subtraction from left to right. In this case, it's \( -2 - (-1) \).

Following these steps carefully is critical to arrive at the correct simplified expression. Remember, disregarding this sequence can lead to incorrect results.

Exponentiation

The process of raising a number to a power is known as exponentiation. In algebra, the exponent tells us how many times to multiply the base by itself. For example, \( a^n \) means that you multiply \( a \) by itself \( n \) times.

In the given exercise, we have the expression \( (-2)^2 \) which implicates exponentiation. Here's what it means:

• \( (-2)^2 \) instructs us to multiply \( -2 \) by itself once: \( -2 \times -2 \).
• A negative multiplied by a negative yields a positive result, so \( -2 \times -2 = 4 \).

The concept of exponentiation can sometimes be challenging since the base and the exponent can be any number or expression. However, always remember the base is what gets multiplied and the exponent tells you how many times the multiplication occurs.

Basic Algebra

At its core, basic algebra involves finding the value of unknowns in equations, but it also encompasses simplifying expressions, which is our focus in this exercise. The key to mastering basic algebra is understanding and correctly applying various mathematical operations such as addition, subtraction, multiplication, division, and exponentiation.

In our example, basic algebra is applied to simplify the numerical expression. The steps include:

• Identifying and operating within the constraints set by the order of operations.
• Understanding the properties of numbers, for instance, that dividing by a negative inverts the sign.
• Recognizing that subtracting a negative is equivalent to adding a positive allows us to rewrite \( -2 - (-1) \) as \( -2 + 1 \).

Solving such problems requires structured thinking and the patience to work through each step methodically. With practice, these fundamental principles become second nature, allowing for quick and accurate simplification of algebraic expressions.