Problem 4 - Precalculus: Functions and Graphs

Question

Express the number in the form \(a / b,\) where \(a\) and \(b\) are integers. $$\frac{2^{0}+0^{2}}{2+0}$$

Step-by-Step Solution

Answer

The number is expressed as \(\frac{1}{2}\).

Step 1: Simplify the Numerator

The numerator of the given expression is \(2^0 + 0^2\). Since any number raised to the power of zero is 1, \(2^0 = 1\). And any number raised to the power of something, when that number is zero, is also zero, so \(0^2 = 0\). Therefore, the simplified numerator is \(1 + 0 = 1\).

Step 2: Simplify the Denominator

The denominator of the expression is \(2 + 0\). Adding these numbers gives us \(2\).

Step 3: Final Fraction

With the simplified numerator and denominator, we can express the number as a fraction. Therefore, the fraction is \(\frac{1}{2}\).

Key Concepts

Expression Simplification

Expression Simplification involves reducing a complex expression to its simplest form, making it easier to understand and work with. In the given exercise, simplifying the expression \[\frac{2^{0}+0^{2}}{2+0}\]requires breaking it down into smaller parts.

• First, we simplified the numerator \(2^0 + 0^2\).
• Then, we simplified the denominator \(2 + 0\).

By solving each component step-by-step:

• The term \(2^0\) evaluates to 1 because any non-zero base raised to the power of zero equals 1.
• Secondly, \(0^2\) simplifies to 0 because zero raised to any power remains zero.
• Adding these gives a numerator of 1.
• The denominator \(2 + 0\) simplifies directly to 2.

Hence, the expression simplifies smoothly to the easily manageable fraction \(\frac{1}{2}\). This methodical approach of simplifying step-by-step helps to avoid errors and confusion.

Exponents

Exponents are a shorthand way to represent repeated multiplication of the same factor. They are written as a number (the base) raised to an exponent (or power). For instance, in the expression \(2^0\), 2 is the base and 0 is the exponent.

• An important exponent rule is that any non-zero number raised to the power of zero equals 1. Therefore, \(2^0 = 1\).
• Another rule is that zero raised to any positive power is zero, reflecting that zero multiplied by itself any number of times will always be zero. Consequently, \(0^2 = 0\).

Understanding these rules allows for quick and accurate simplification of expressions involving exponents. This knowledge significantly facilitates the process of expression simplification, as seen in the original exercise's numerator simplification.

Integer Arithmetic

Integer Arithmetic focuses on performing operations like addition, subtraction, multiplication, and division on integers, which are whole numbers. In the exercise, Integer Arithmetic was used to simplify both the numerator and the denominator.

• The numerator involved the addition of two integer values, \(1 + 0\), which results in an integer, 1.
• The denominator consisted of the addition, \(2 + 0\), leading to the integer 2.

By understanding Integer Arithmetic, one can easily calculate and combine integer values, leading to a simplified representation of more complex fractions. This concept is crucial in expressing fractions accurately, as seen where the final result of the given exercise is a clean fraction \(\frac{1}{2}\). Knowing how to manipulate integers correctly ensures the precision and clarity of any mathematical solution.