Question

For Exercises \(1-74,\) simplify. $$\frac{3 \cdot(-6)-6^{2}}{-2(2-5)^{2}}$$

Short answer

Question: Simplify the complex fraction $\frac{3\cdot(-6) - 6^{2} }{-2(2-5)^{2}}$. Answer: 3

Step-by-step solution

Part 1: Numerator

First of all, let's focus on the numerator. We need to perform the operations given in the right order, which is multiplication first and then subtraction: $$ 3 \cdot(-6) - 6^{2} = -18 - 36 $$ Now, we need to perform the subtraction: $$ -18 - 36 = -54 $$

Part 2: Denominator

Now, let's look at the denominator. We need to deal with the operations in the order of parentheses, exponentiation, and multiplication: $$ -2(2-5)^{2} = -2(-3)^{2} $$ Now, we deal with the exponentiation: $$ -2 \cdot 9 $$ Finally, we perform the multiplication: $$ -2 \cdot 9 = -18 $$

Part 3: Simplification

Now, we have the simplified numerator and denominator: $$ \frac{-54}{-18} $$ To simplify the fraction, we will divide both the numerator and the denominator by their GCD, which is 18: $$ \frac{-54 / 18 }{-18 / 18} = \frac{3}{1} $$ The simplified fraction is simply 3, as dividing by 1 does not change it. Therefore, the simplified expression is: $$ \boxed{3} $$

Key concepts

Simplification

Simplification is the process of making an expression easier to work with or understand by reducing its complexity. In mathematics, this often means combining like terms, factoring, or applying arithmetic operations. In the context of fractions, simplification involves reducing a fraction to its simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For example, when you encounter a fraction like \(\frac{-54}{-18}\), you first look for the GCD of 54 and 18, which is 18. You then divide both numbers by 18 to get the simplified fraction \(\frac{3}{1}\). Once simplified, the fraction represents the simplest equivalent value, which is just 3.

• Simplification reduces errors by making an expression less cumbersome.
• It helps in seeing relationships and patterns more clearly.
• Ensures numerical results are presented in their most basic form.

Understanding simplification is crucial in prealgebra because it lays the groundwork for more complex problem-solving.

Order of Operations

Understanding the order of operations is crucial when simplifying mathematical expressions. It's the sequence in which operations such as addition, subtraction, multiplication, and division are performed to ensure consistent and accurate results.In mathematics, the order of operations is often remembered using the acronym PEMDAS:

• Parentheses – Begin with calculations inside parentheses.
• Exponents – Next, calculate powers or roots.
• Multiplication and Division – From left to right, perform these operations as they appear.
• Addition and Subtraction – Finally, proceed with these operations from left to right.

In the provided exercise, the order of operations was applied as follows:

• First, solve the parentheses and the exponents in the denominator: \((-3)^2\) becomes 9.
• Perform multiplication: Iterate through operations within parentheses, then multiply by coefficients: \(-2 \cdot 9 = -18\).
• Solve the entire expression by addressing the numerator \(3 \cdot (-6) - 6^2 = -54\).

Following these steps ensures that you tackle each part of an expression systematically, leading to correct results every time.

Fractions

Fractions are numerical quantities that express a part of a whole. They consist of a numerator and a denominator, which divides the numerator. In prealgebra, understanding fractions is pivotal as they are ubiquitous in mathematics.A fraction like \(\frac{-54}{-18}\) can be simplified by finding a common factor between both numbers. Reducing a fraction involves dividing both terms by their greatest common divisor.To simplify:

• Identify the GCD of the numerator and denominator.
• Divide both by this number. For example, dividing -54 and -18 by 18 simplifies the faction to \(\frac{3}{1}\).

Fractions can sometimes appear in complex problems where addition, subtraction, multiplication, or division between fractions is required. Becoming comfortable handling fractions involves practice with fraction arithmetic and simplification techniques. This ability enables you to convert improper fractions to mixed numbers, simplify complex fractions, and solve real-world problems involving parts of quantities.