Question

For Exercises \(1-74,\) simplify. $$\frac{(-2)^{3}+5(7-15)}{4 \cdot 11-2(5-9)^{2}}$$

Short answer

Question: Simplify the given expression: $$\frac{7 - 15 -(5*(-2)^{3})}{(5 - 9)(4*11 - (-2*(-4)^{2}))}$$ Answer: The simplified expression is \(\frac{-12}{19}\).

Step-by-step solution

Calculate Expressions within Parentheses

First, we'll simplify the expressions within parentheses in the numerator and the denominator: Numerator: \(7-15=-8\) Denominator: \(5-9=-4\)

Apply Exponents

Next, we'll calculate the exponent in the numerator: \((-2)^{3} = -8\)

Perform Multiplication and Division

In the numerator, we have a multiplication: \(5*(-8) = -40\). Now, we will calculate the results of the two multiplications in the denominator: \(4*11=44\) and \(-2*(-4)^{2}=-2*16=-32\).

Perform Addition and Subtraction

Now, we can perform the addition and subtraction operations: Numerator: \((-8) + (-40) = -48\) Denominator: \(44 - (-32) = 44 + 32 = 76\)

Express the Simplified Fraction

Now that we have simplified both the numerator and the denominator, we can write the result as a single fraction: $$\frac{-48}{76}$$

Simplify the Fraction

The greatest common divisor (GCD) of 48 and 76 is 4. To simplify the fraction, we'll divide both the numerator and the denominator by the GCD: $$\frac{-48 ÷ 4}{76 ÷ 4} = \frac{-12}{19}$$ The simplified expression for the given problem is \(\frac{-12}{19}\).

Key concepts

Order of Operations

In math, it is crucial to follow the correct sequence when solving expressions. This is known as the "Order of Operations." It ensures that everyone calculates expressions in a consistent manner, yielding the same result. The common acronym for remembering the order is PEMDAS:

• Parentheses: Always start with calculations within parentheses.
• Exponents: Next, solve any exponents or powers.
• Multiplication and Division: From left to right, solve multiplication and division.
• Addition and Subtraction: Finally, perform all addition and subtraction from left to right.

When simplifying expressions such as \[\frac{(-2)^{3}+5(7-15)}{4 \cdot 11-2(5-9)^{2}}\]it's important to follow PEMDAS to avoid mistakes. Start with simplifying the expressions inside the parentheses, then move to exponents. Next, handle the multiplication and division, and finally, the addition and subtraction.

Fractions

Fractions represent parts of a whole. They consist of two main parts: a numerator, the number above the line, and a denominator, the number below the line. When simplifying expressions that result in fractions, like \[\frac{-48}{76}\], using fractions helps in breaking down a complex problem into simpler parts.
Fractions are commonly used to express numbers that are not whole, making them incredibly useful for precise calculations. Knowing how to manipulate fractions by finding common denominators and equivalent fractions is essential for solving problems accurately.
In our problem, after performing all operations, we ended up with the fraction \[\frac{-48}{76}\]. This required further simplification using the greatest common divisor, which we will discuss next.

Greatest Common Divisor (GCD)

The concept of the greatest common divisor (GCD) is important when simplifying fractions. The GCD is the largest number that can evenly divide two or more numbers. To simplify a fraction, we divide both the numerator and the denominator by their GCD.
In our exercise, we needed to simplify the fraction \[\frac{-48}{76}\]. By identifying that the GCD of 48 and 76 is 4, we divide both the numerator and the denominator by 4:\[\frac{-48 \div 4}{76 \div 4} = \frac{-12}{19}\]By doing so, we have simplified the fraction to its most reduced form. Knowing how to find and apply the GCD is essential in reducing fractions efficiently.

Numerators and Denominators

Understanding numerators and denominators is key to working with fractions. The numerator is the top number in a fraction and represents how many parts we are considering. The denominator, on the bottom, indicates the total number of equal parts.

• Numerator: In our simplified fraction \(\frac{-12}{19}\), the (-12) is the part we focus on.
• Denominator: The (19) represents the whole divided into equal parts.

This understanding aids in the manipulation of fractions, allowing us to perform operations like addition, subtraction, multiplication, and division. By examining the numerical relationship between the numerator and the denominator, we can easily simplify or compare different fractions. Always remember to analyze both components to achieve a clearer mathematical expression.