Problem 22 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The value of the expression is 3.

Step 1: Understand Exponents

First, it's important to understand that any number raised to the power of 0 is equal to 1. This means that both \(x^0\) and \((4x)^0\) are equal to 1.

Step 2: Simplify the Numerator

Look at the numerator \(3x^0\). Since \(x^0 = 1\), this simplifies to \(3 \times 1 = 3\).

Step 3: Simplify the Denominator

Now consider the denominator \((4x)^0\). Since anything raised to the power of 0 is 1, \((4x)^0 = 1\).

Step 4: Calculate the Fraction

Now substitute the simplified numerator and denominator into the expression: \(\frac{3}{1}\). This simplifies to just 3.

Key Concepts

Zero Exponent Rule

When dealing with exponents, one of the most crucial rules to grasp is the zero exponent rule. This rule states that any non-zero base raised to the power of zero equals 1. In mathematical terms, this can be expressed as:
\( a^0 = 1 \)
where \( a \) is any non-zero number. This might seem a bit counterintuitive at first, but it is a foundational part of exponent rules.To see this in action, consider our expression: \( x^0 \) and \( (4x)^0 \). Regardless of what value \( x \) holds (as long as it isn't zero), both \( x^0 \) and \( (4x)^0 \) are 1. The zero exponent simplifies expressions by reducing complex terms to a single, manageable form. This understanding becomes particularly helpful when simplifying fractions or expressions with multiple levels of complexity.

Numerator and Denominator Simplification

Simplifying expressions with fractions involves breaking them down into simpler parts. Let's start with the numerator. In the expression \( \frac{3x^0}{(4x)^0} \), notice the term \( 3x^0 \) in the numerator. Using the zero exponent rule, we know that \( x^0 \) simplifies to 1. Thus, the numerator becomes:

• \(3x^0 = 3 \times 1 = 3\)

Next, examine the denominator, \((4x)^0\). Applying the zero exponent rule here simplifies the denominator to 1:

• \((4x)^0 = 1\)

This turns our fraction into a much simpler form, making it easier to evaluate and understand.

Expression Evaluation

With both the numerator and denominator simplified, it's time to perform the final expression evaluation. Our task is now to evaluate \( \frac{3}{1} \). This is straightforward, as dividing by 1 doesn't change the value of the numerator. So, the expression evaluates directly to:

• \(3 / 1 = 3\)

Breaking down the expression into its components and understanding each part with the help of rules like the zero exponent rule, you can simplify seemingly complex expressions with confidence. This process not only speeds up calculations but also reinforces the fundamental principles of algebra. Always remember, simplifying expressions can make challenging problems become more manageable.