Problem 5 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The number is expressed as \(\frac{-47}{3}\).

Step 1: Expand and Calculate Exponents

First, calculate each part of the expression separately. The expression is \(-2^4 + 3^{-1}\).Calculate \(2^4\): \(2^4 = 2 \times 2 \times 2 \times 2 = 16\). Therefore, \(-2^4\) equals \(-16\). Next, calculate \(3^{-1}\). Recall that a negative exponent indicates a reciprocal: \(3^{-1} = \frac{1}{3}\).

Step 2: Add the Results

Now, add the two values obtained: \(-16\) and \(\frac{1}{3}\). The expression \(-16 + \frac{1}{3}\) requires a common denominator to combine these into a single fraction.

Step 3: Find a Common Denominator

The integer \(-16\) can be represented with the same denominator as \(\frac{1}{3}\), which means expressing \(-16\) in terms of a denominator of \(3\):\(-16 = \frac{-16 \times 3}{3} = \frac{-48}{3}\).

Step 4: Combine Fractions

Add the fractions: \(\frac{-48}{3} + \frac{1}{3} = \frac{-48 + 1}{3}\). This simplifies to: \(\frac{-47}{3}\).

Step 5: Write the Final Answer

Thus, when the number is expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are integers, it results in \(\frac{-47}{3}\).

Key Concepts

Integer

An integer is a complete or whole number that can be positive, negative, or zero. In mathematics, integers are essential building blocks. They do not have fractions or decimals. You can think of them as the numbers you count, if you include both positive and negative numbers along with zero.
Examples of integers are:

• -2, 0, 5, and 42

When dealing with exercises involving expressions like \(-2^4 + 3^{-1}\), recognizing which parts of the expression are integers is crucial. Here, \(-2^4\) initially evaluates to an integer; it results in \(-16\) once calculated. Distinguishing integers helps us to apply proper mathematical operations, particularly when combining different types of numbers.

Negative Exponent

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. Understanding this concept is fundamental in simplifying expressions which involve exponents. For instance, with \(3^{-1}\), it signifies the reciprocal: \(\frac{1}{3}\).
This conversion is crucial for simplifying and combining terms in an expression. Here are key takeaways about negative exponents:

• For any non-zero number \(a\), \(a^{-n} = \frac{1}{a^n}\)
• Negative exponents represent division rather than multiplication.

When simplifying expressions with both positive and negative exponents, manage each term individually before combining them, like in \(-2^4 + 3^{-1}\). Recognizing the reciprocal of the negative exponent ensures accurate calculations.

Common Denominator

Finding a common denominator is an important step when adding or subtracting fractions. This process ensures that fractions are expressed in a way that makes them easy to combine. In our exercise, we need to add \(-16\) and \(\frac{1}{3}\). First, convert \(-16\) to a fraction with a denominator of 3, making it \(\frac{-48}{3}\).
Here are practical tips to find a common denominator:

• Identify the least common multiple (LCM) of the denominators.
• Adjust each fraction to have the common denominator.

This enables straightforward addition or subtraction of fractions. Simply add or subtract the numerators while keeping the common denominator, like combining \(\frac{-48}{3} + \frac{1}{3}\), resulting in \(\frac{-47}{3}\). Mastering the technique of finding a common denominator is essential for simplifying expressions involving fractions.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. Once you have a fraction, like \(\frac{-47}{3}\), check if the numerator and denominator can be divided by the same number other than 1.
Steps for simplifying fractions include:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.

In our case, \(-47\) and \(3\) have no common factors other than 1, so \(\frac{-47}{3}\) is already in its simplest form. Simplifying fractions makes them easier to interpret and use in calculations. It is also the final step in ensuring your answer is presented in a standardized way, useful for both understanding and communication of mathematical concepts.