Problem 21 - Algebra 2 and Trigonometry

Question

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ \frac{3^{0}}{4^{0}} $$

Step-by-Step Solution

Answer

The expression \(\frac{3^0}{4^0}\) evaluates to 1.

Step 1: Identify the Problem

The exercise involves evaluating the expression \( \frac{3^0}{4^0} \) when \( x eq 0 \). This step confirms that since there is no \( x \) explicitly in the expression, the condition \( x eq 0 \) does not directly affect this expression.

Step 2: Evaluate Exponents

Start by recalling the rule for any non-zero number raised to the power of 0, which states that \( a^0 = 1 \). This rule applies because any number to the power of zero is considered to be 1 by definition.

Step 3: Simplify the Expression

Using the rule from Step 2, convert both numerators and denominators to 1: \( 3^0 = 1 \) and \( 4^0 = 1 \). Therefore, the expression becomes \( \frac{1}{1} \).

Step 4: Compute the Final Result

Since \(\frac{1}{1}\) simplifies to 1, this is our final result of the expression.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. It's similar to fractions in arithmetic, but instead of numbers, you're dealing with algebraic expressions.
An example is \( \frac{x+2}{y-3} \), where \( x+2 \) is the numerator and \( y-3 \) is the denominator.
When working with rational expressions:

• Simplify by factoring both the numerator and the denominator.
• Cancel common factors between the numerator and the denominator.
• Remember to consider restrictions, which are values that make the denominator zero, as they are not allowed.

Understanding rational expressions is crucial for solving various types of algebra problems, including those involving exponents, equations, and inequalities.

Simplifying Expressions

Simplifying an expression means reducing it to its simplest form, where it cannot be simplified further. This often involves combining like terms, factoring, and using mathematical identities.
For example, if you encounter \( \frac{3^0}{4^0} \):

• Recall that \( 3^0 = 1 \) and \( 4^0 = 1 \) due to the zero exponent rule.
• The expression then becomes \( \frac{1}{1} \), which simplifies to 1.

So, by applying the zero exponent rule, we simplify complex expressions easily. When simplified, expressions are typically easier to understand and work with in subsequent calculations.

Basic Exponents

Exponents are a fundamental component of algebra. They indicate how many times a number, known as the base, is multiplied by itself.
Starting with the basics:

• \( a^n \) means the base \( a \) is used as a factor \( n \) times.
• When \( n = 0 \), by definition, \( a^0 = 1 \) for any non-zero \( a \). This is known as the zero exponent rule.

Understanding basic exponents is critical as they are foundational in manipulating and simplifying more complex algebraic expressions. When learning about exponents, keep these tips in mind:

• Practice the power rules: \((a^m)^n = a^{mn}\) and \(a^m \times a^n = a^{m+n}\).
• Recognize special cases like \(a^0 = 1\) to simplify expressions quickly.

Mastering these basics will aid you in tackling more advanced problems, including those involving rational expressions.