Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-10: Express the number in the form \(a / b\), where \(a\) and \(b\) are integers. $$ -2^{4}+3^{-1} $$

Step-by-Step Solution

Answer

\(-2^4 + 3^{-1} = \frac{-47}{3}\).

Step 1: Evaluate the Powers

First, evaluate the individual powers in the expression. The expression given is \[-2^4 + 3^{-1}\]. We evaluate each part separately:1. \(2^4 = 16\). Since there is a negative sign before this term, \(-2^4 = -16\).2. \(3^{-1} = \frac{1}{3}\).

Step 2: Combine the Terms

Now combine the terms. We substitute the evaluated powers back into the expression:\[-16 + \frac{1}{3}\].

Step 3: Express with a Common Denominator

To combine \(-16\) and \(\frac{1}{3}\), convert \(-16\) to a fraction with a denominator of 3:\[-16 = \frac{-16 imes 3}{3} = \frac{-48}{3}\].

Step 4: Add the Fractions

Now add the two fractions:\[\frac{-48}{3} + \frac{1}{3} = \frac{-48 + 1}{3} = \frac{-47}{3}\].

Step 5: Write the Final Fraction

The expression \(-2^4 + 3^{-1}\) can be expressed in the form \(\frac{a}{b}\) as:\[\frac{-47}{3}\].

Key Concepts

Arithmetic Operations

Arithmetic operations encompass basic mathematical functions like addition, subtraction, multiplication, and division. They are fundamental in simplifying complex expressions, such as the original exercise. Each operation serves a distinct purpose:

• Addition and Subtraction: Used to combine or remove values respectively. In our example, we added and subtracted numbers to simplify the expression.
• Multiplication and Division: Involves scaling numbers up or down. Here, division helps express values as fractions.

Performing arithmetic operations in the correct sequence, following the order of operations (PEMDAS/BODMAS), is crucial for obtaining the right answer. This means powers and roots are solved before combining terms.

Powers and Roots

Powers, such as squaring or taking higher exponents, increase numbers exponentially. In the example, we calculate the power of 2, specifically \(2^4 = 16\).

• A base raised to an exponent such as \(a^n\) means multiplying the base \(a\) by itself \(n\) times.
• In the exercise, \(-2^4\) is equivalent to \(-16\) using the rule that negative outside an exponent affects the result directly.

Roots, although not used directly in this problem, are closely linked to powers as the inverse operation. They are key for expressing numbers in simplified forms when dealing with square roots or other root terms.

Fraction Addition

Adding fractions requires a common denominator. This ensures the fractions are comparable and can be added together seamlessly. In our exercise, we have \(-16\) and \(\frac{1}{3}\).

• First, transform \(-16\) to a fraction with the same denominator as \(\frac{1}{3}\), resulting in \(\frac{-48}{3}\).
• Then, add \(\frac{-48}{3}\) and \(\frac{1}{3}\).This sum is straightforward: you add the numerators while keeping the denominator the same.
• The result of \(\frac{-48}{3} + \frac{1}{3}\) is \(\frac{-47}{3}\).

Understanding this method is vital for handling fractional arithmetic without errors.

Numerical Expressions

Numerical expressions are combinations of numbers, operations, and occasionally variables that represent a particular value. The original exercise: \(-2^4 + 3^{-1}\) exemplifies such an expression, requiring careful simplification.

• Start by resolving parts within the expression, like powers, before moving to simpler arithmetic.
• Each term is evaluated separately then combined into a single simplified form.
• The expression eventually translates to \(\frac{-47}{3}\), a format showing clear integer relationships.

By breaking down numerical expressions step-by-step, complex calculations become approachable and manageable.