Problem 114 - Introductory Algebra for College Students

Question

Simplify: \(\frac{1}{4}-6(2+8) \div\left(-\frac{1}{3}\right)\left(-\frac{1}{9}\right)\)

Step-by-Step Solution

Answer

-1619.75

Step 1: Simplify the expression inside the parentheses

First, we need to simplify the expression inside the parentheses, which is \(2 + 8\). This results in \(2 + 8 = 10\).

Step 2: Simplify the expression with the division and fractions

Next, we see a division operation involving fractions. Recall that dividing by a fraction is the same as multiplying by its reciprocal. Here we have \(10 ÷ \left(-\frac{1}{3}\right)\left(-\frac{1}{9}\right)\), which rewrites to \(10 × \left(-3\right)\left(-9\right)\). This results in \(10 × 27 = 270\).

Step 3: Simplify the expression with multiplication

After this, we have \(6 × 270\), it simplifies to 1620.

Step 4: Simplify the expression with the subtraction

Lastly, we have a subtraction operation left, \( \frac{1}{4} - 1620\). It simplifies to \(-1619.75\).

Key Concepts

Understanding the Order of Operations

When tackling algebraic expressions, it's crucial to follow the order of operations to simplify expressions correctly. The common mnemonic is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This helps organize the process of solving problems:

• Begin by simplifying expressions inside the Parentheses.
• Next, tackle any Exponents.
• Then, execute any Multiplication or Division operations, moving left to right.
• Finally, handle Addition and Subtraction in a left-to-right sequence.

In our original exercise, you start by simplifying the expression within the parentheses, which is the addition of numbers 2 and 8, followed by handling the division and multiplication operations. Ensure each step follows the sequence to avoid errors.

Working with Fractions

Fractions represent parts of a whole and can complicate expressions if not handled properly. Key to simplifying expressions with fractions is understanding how to divide and multiply them effectively.

• To divide by a fraction, multiply by its reciprocal. This flips the numerator and the denominator of the fraction. For example, dividing by \(-\frac{1}{3}\) becomes multiplying by \(-3\).
• Multiplying fractions is straightforward: multiply the numerators together and the denominators together. When dealing with whole numbers, they can be considered as fractions with a denominator of 1.

In the exercise, using this method allows us to convert a complex division involving fractions into a simpler multiplication problem, facilitating further simplification.

Performing Mathematical Operations

Mathematical operations include addition, subtraction, multiplication, and division. Each has its rules, especially when involving different elements like fractions, negative numbers, or large numbers.

• Multiplication involves combining quantities in groups, such as when distributing over addition or simplifying expressions.
• Subtraction represents taking away quantities, which can introduce negative results if the subtrahend exceeds the minuend.
• Division breaks quantities into equal parts, reversed when dealing with fractional divisors by using multiplication of the reciprocal.

In the given problem, multiplication is pivotal in converting fractions into manageable integers, and subtraction finalizes the result, demonstrating its simple yet essential role in simplifying complex expressions.

Handling Negative Numbers

Negative numbers add another layer of complexity to solving mathematical expressions, but with careful attention, they can be managed effectively.

• When multiplying or dividing two negative numbers, the result is positive. In contrast, multiplying or dividing a positive number by a negative number results in a negative.
• Subtraction involving negative numbers requires additional care: subtracting a negative number is equivalent to adding its positive counterpart.

The original exercise effectively demonstrates these rules with \((10 \times -3 \times -9)\), where the double negative results in positive 27, which simplifies further arithmetic operations. Understanding these basics ensures we're not derailed by the presence of negative values when simplifying expressions.