Problem 63 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{16 x^{4}}{32 x^{8}} $$

Step-by-Step Solution

Answer

The simplified expression is \(0.5x^{-4}\)

Step 1: Simplify Coefficients

Start the simplification by handling the numerical coefficients, which are 16 and 32. As both numbers are divisible by 16, simplifying gives \(\frac{16}{32} = 0.5\)

Step 2: Simplify Variable

Next, simplify the variable which is \(x^{4}\) and \(x^{8}\). According to the power division rule (if bases are equal, then subtract the exponent in the denominator from the exponent in the numerator), we have \(x^{4-8} = x^{-4}\)

Step 3: Combine Simplified Coefficients and Variable

Combine the results from step 1 and 2 for the final simplified expression. Multiplying together 0.5 and \(x^{-4}\) returns \(0.5x^{-4}\)

Key Concepts

Numerical Coefficient Simplification

When encountering algebraic expressions like \(\frac{16 x^{4}}{32 x^{8}}\), simplifying numerical coefficients should usually be your first step. A numerical coefficient is simply a numerical factor multiplied by a variable. To simplify, look for common factors in the coefficients. In this case, both 16 and 32 can be divided by 16. This process is akin to reducing a fraction to its simplest form.

For example, with \(\frac{16}{32}\), you can divide both the numerator and the denominator by their greatest common divisor, which is 16. This simplifies \(\frac{16}{32}\) to \(\frac{1}{2}\) or 0.5. Simplifying coefficients makes the rest of the process more manageable.

Variable Exponent Rules

Variables raised to an exponent, like \(x^{4}\) and \(x^{8}\), are central to understanding algebraic expressions. When variables of the same base are divided, you will apply the variable exponent rules, specifically the power division rule. This entails subtracting the exponent of the denominator from the exponent of the numerator if the bases are the same.

Using our example, \(x^{4}\) divided by \(x^{8}\) becomes \(x^{4-8}\) or \(x^{-4}\). These rules are handy tools used to simplify expressions into their most compact form. Remember, when the exponent is negative, it indicates the reciprocal of the base raised to the corresponding positive exponent.

Power Division Rule

The power division rule is a key concept in algebra that guides us when dividing terms with exponents. In our expression \(\frac{x^{4}}{x^{8}}\), since the bases (x) are identical, you can subtract the exponents. Subtracting the bottom exponent (8) from the top exponent (4), you get \(x^{4-8}\) or \(x^{-4}\).

However, what does a negative exponent mean? If you have a negative exponent, it translates to one over the base with a positive exponent. In other words, \(x^{-4}\) can be written as \(\frac{1}{x^{4}}\). The power division rule streamlines the process of simplifying expressions, often resulting in a more digestible and easier to work with form.