Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-10: 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 expression simplifies to \(\frac{1}{2}\).

Step 1: Simplify the Numerator

Evaluate the expression in the numerator \(2^0 + 0^2\). First, calculate \(2^0\), which is 1, since any number to the power of 0 is 1. Then, calculate \(0^2\), which is 0. Adding these results, the numerator is \(1 + 0 = 1\).

Step 2: Simplify the Denominator

Evaluate the expression in the denominator \(2 + 0\). This simplifies directly to \(2\), since adding 0 does not change the value.

Step 3: Form the Fraction

Now that both the numerator and the denominator are simplified, form the fraction. Place the numerator 1 over the denominator 2, resulting in the fraction \(\frac{1}{2}\).

Step 4: Check and Confirm Integer Values

Ensure that both \(a\) and \(b\) in the fraction \(\frac{a}{b}\) are integers. Here, \(a = 1\) and \(b = 2\), both of which are integers. Therefore, the expression \(\frac{1}{2}\) meets the requirement of the form \(\frac{a}{b}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra. It involves breaking down complex mathematical phrases into their simplest form. In our exercise, the expression in the numerator, \(2^0 + 0^2\), was simplified by individually assessing each part of the expression.

When simplifying:

• Evaluate each component separately. Here, \(2^0\) equals 1 because any number raised to the power of zero is always 1. Similarly, \(0^2\) equals 0 since zero raised to any power remains zero.
• Combine the simplified values through addition or subtraction. For our expression, adding the results of \(1 + 0\) gives 1.
• Simplify the denominator in a similar fashion. In this exercise, the denominator is \(2 + 0\), which directly simplifies to 2 since adding zero does not change the original value.

Once you simplify both the numerator and denominator, you can express them as a fraction that shows the relationship between the two values. This process ensures clarity and precision in mathematical expressions.

Integer Arithmetic

Integer arithmetic refers to mathematical operations involving whole numbers, which can be positive, negative, or zero. In our problem, integer arithmetic plays a crucial role when assessing both the numerator and the denominator of the given expression.

Important aspects of integer arithmetic include:

• Addition: Adding integers involves combining their values. In our exercise, after finding \(2^0 = 1\) and \(0^2 = 0\), we added them to get 1. The principle is straightforward: the sum of any number and zero is the number itself.
• Subtraction: Although not directly used in this example, it's important to remember that subtraction of integers simply involves finding the difference.
• Order of operations: Always respect the mathematical hierarchy (parentheses, exponents, multiplication and division, addition and subtraction), which ensures that calculations are performed in the correct sequence.

Understanding how to perform these operations accurately is essential. It helps to simplify expressions with confidence and ensures the results make mathematical sense.

Exponents

Exponents are a mathematical way to represent repeated multiplication. They are a crucial part of algebra and were used to simplify the expression in this exercise.

Key points about exponents:

• Meaning: An expression like \(2^0\) specifies that 2 is raised to the power of 0. A fundamental rule is that any non-zero number to the power of zero equals 1.
• Zero exponent rule: This rule states that no matter the base (as long as it isn’t zero), raising it to the power of zero results in 1. Hence, \(2^0 = 1\).
• Positive exponents: Represent repeated multiplication, such as \(3^2\) which means \(3 \times 3 = 9\).

In our problem, understanding exponents helps to correctly evaluate and simplify the numerator. With these basics, students can tackle more complex problems involving exponents in the future.