Problem 72 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ \frac{3\left(-3^{2}+2 \cdot 2^{2}\right)}{(5-8)(7-9)} $$

Step-by-Step Solution

Answer

\(-\frac{1}{2}\)

Step 1: Simplify the Expression Inside the Numerator Parentheses

First, simplify the terms inside the numerator's parentheses. Evaluate \(-3^2\) and \(2 \cdot 2^2\).\(-3^2\) simplifies to \(-9\) because the exponentiation comes before the negation.\(2 \cdot 2^2 = 2 \cdot 4 = 8\).

Step 2: Calculate the Result of the Numerator

Now substitute the simplified terms back into the numerator:\[3\left(-9 + 8\right) = 3(-1) = -3\]

Step 3: Simplify the Expression Inside the Denominator Parentheses

Simplify the terms inside the denominator's parentheses:\(5 - 8 = -3\)\(7 - 9 = -2\)

Step 4: Calculate the Result of the Denominator

Now multiply the simplified terms:\((-3)(-2) = 6\)

Step 5: Divide the Simplified Numerator by the Simplified Denominator

Finally, divide the result from Step 2 by the result from Step 4:\(\frac{-3}{6} = -\frac{1}{2}\)

Key Concepts

Exponentiation

Exponentiation involves raising a number, known as the base, to the power of an exponent. In this operation, the base is multiplied by itself as many times as indicated by the exponent. For example, in the expression \( -3^2 \), you might be tempted to think of it as \( (-3) \times (-3) \). However, the square only applies to the 3, which then becomes 9, before the negative is applied, resulting in \( -9 \). Remember:

• Exponents are performed before any other operations, according to the "PEMDAS/BODMAS" rule (Parentheses/Brackets, Exponents/Orders, Multiplication-Division, Addition-Subtraction).
• The exponent only affects what is immediately to its left, unless the base is enclosed in parentheses.

This subtle distinction often causes confusion, so it's important to carefully consider how exponents apply to negative numbers.

Numerator and Denominator

In a fraction, the top number is called the numerator and the bottom number is the denominator. These two components are crucial in determining the value of a fraction. For instance, in the expression \( \frac{3(-9 + 8)}{(5-8)(7-9)} \),

• The result of the operations resolved within the numerator gives us \(3(-1)\), which equals \(-3\).
• Similarly, the denominator simplifies within its own parentheses to \( (-3)(-2) = 6 \).

Fractions serve as division problems, where the numerator is divided by the denominator. When solving, be sure to simplify both separately before performing the division. Understanding this helps to prevent errors and ensures clarity in complex mathematical expressions.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. This process requires a logical approach using the order of operations. In the given problem, you solve for expressions inside parentheses first, then move through other operations:

• For the numerator, resolve \( -3^2 \) to \(-9\) and \( 2 \times 2^2 \) to \( 8 \). Combine these in the parentheses to get \( -1 \), allowing the numerator to simplify to \( -3 \).
• In the denominator, handle the expressions \( 5-8 \) and \( 7-9 \) separately, yielding \( -3 \) and \( -2 \), respectively. These combine and multiply to form \( 6 \).

Simplifying expressions requires patience and attention to detail. Start with the elements inside the inner layers of parentheses and work outward. Each step should follow the hierarchy of operations, making sure no step is skipped or performed prematurely. This ensures accuracy and confidence in your final results.