Problem 20 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The value of the expression is \( 0.25 \).

Step 1: Understand the Expression

We are given the expression \( \frac{3^0}{4} \) and asked to find its value when \( x eq 0 \). However, this expression does not involve \( x \), so we just evaluate the expression itself.

Step 2: Evaluate the Exponent

Recall that any number raised to the power of zero equals 1, provided the base is not zero. Therefore, \( 3^0 = 1 \).

Step 3: Simplify the Fraction

Substitute the value found in Step 2 into the expression: \( \frac{1}{4} = 0.25 \). This fraction represents a quarter.

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. Understanding how to work with exponents is crucial for simplifying algebraic expressions. When an exponent is zero, it may seem unclear at first what this means. The key rule to remember is that any non-zero number raised to the power of 0 is 1. This is because exponents represent how many times to multiply the number by itself; with an exponent of zero, it means that you don't actually multiply it at all. For example, in our expression with base 3, we have:

• If we have any number to the power of 0, such as 3 raised to the 0, it simplifies to 1, provided the base isn't zero: \(3^0 = 1\).
• Mathematically this is often justified because it maintains consistency with the rules of exponents; if the exponent decreases by 1, you divide by the base number.

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its simplest or most basic form. This makes mathematics easier because we work with smaller numbers. A fraction is simplified when the numerator and the denominator have no common factors other than 1. In our example, \(\frac{3^0}{4}\), once we've established that \(3^0\) is equal to 1, we can rewrite this as: \[ \frac{1}{4} \] Let's understand why:

• The numerator, 1, and the denominator, 4, do not have any common factors other than 1.
• Thus, it is already in its simplest form as it cannot be reduced further.

Knowing fractions in their simplest form helps in further mathematical calculations and provides a clearer understanding of proportions.

Fraction Evaluation

When it comes to evaluating fractions, it is simply determining their numerical value. This is especially important when you need a decimal or percent understanding of a particular fraction. For the fraction \(\frac{1}{4}\), let's see how we evaluate it:

• Divide the numerator by the denominator: \(1 \div 4 = 0.25\).
• This tells us that \(\frac{1}{4}\) is equivalent to 0.25 in decimal form.
• Understanding this decimal representation can often provide a intuitively clear picture of the fraction, like knowing that 0.25 represents a quarter, or one-fourth of a whole.

Being comfortable with moving between fractions and decimals is fundamental to many areas of mathematics and real-world applications.