Problem 76 - Elementary and Intermediate Algebra

Question

Perform each division. $$ \frac{112 u z^{4}}{-42 u^{3} z^{8}} $$

Step-by-Step Solution

Answer

The simplified form is \(-\frac{8}{3} u^{-2} z^{-4}\).

Step 1: Identify the Components of the Division

We are given the fraction \( \frac{112 u z^4}{-42 u^3 z^8} \). In this fraction, the numerator is \(112 u z^4\) and the denominator is \(-42 u^3 z^8\). Our task is to simplify this fraction by performing division.

Step 2: Simplify the Coefficients

Divide the coefficients in the numerator and the denominator, which are 112 and -42, respectively. This gives \(\frac{112}{-42} = \frac{-16}{6} = -\frac{8}{3}\).

Step 3: Simplify the Variable 'u'

In the fraction, \(u\) has an exponent of 1 in the numerator and 3 in the denominator. Simplifying \(\frac{u}{u^3}\) leads to \(u^{1-3} = u^{-2}\). This means we will have \(u^{-2}\) in the simplified expression.

Step 4: Simplify the Variable 'z'

The variable \(z\) has an exponent of 4 in the numerator and 8 in the denominator. Simplifying \(\frac{z^4}{z^8}\) results in \(z^{4-8} = z^{-4}\). Thus, we have \(z^{-4}\) in the simplified expression.

Step 5: Combine the Simplified Parts

Bringing together all the simplified components, we arrive at the expression \(-\frac{8}{3} u^{-2} z^{-4}\).

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means finding a fraction that expresses the same value as the original but with the smallest possible numerator and denominator. In algebra, fractions often contain variables and coefficients. Simplification starts with dividing the coefficients – the numerical part – as it forms the basis of the fraction.

In this problem, the numerator is 112 and the denominator is -42. By dividing both by their greatest common divisor, which is 14, we simplify the fraction to \(-\frac{8}{3}\). This creates a simpler fraction that communicates the same ratio or value. However, that's just the beginning! We must also tackle any variables present, using exponent rules.

Exponent Rules

Exponent rules are fundamental in algebra, especially when simplifying expressions with variables. They help in managing powers of the same base easily. When dividing variables with exponents, like in our exercise, we apply the rule:

• If you divide like bases, you subtract the exponents: \( a^{m-n} \).

This means subtracting the exponent of the denominator from that of the numerator for same variables.

For example, in \( u \) with exponents 1 in the numerator and 3 in the denominator, we get \( u^{1-3} = u^{-2} \). For \( z \), which has exponents 4 and 8, \( z^{4-8} = z^{-4} \). Negative exponents signify a reciprocal, but for simplicity, we maintain them as they are still a clear indication of division in the algebraic expression.

Variable Division

Variable division is about simplifying expressions that contain variables with powers. It's important to recognize the rules of operation that each variable follows.

When simplifying variables in a division problem, use the exponent rules to change complex expressions filled with variables into simpler ones. This yields more straightforward forms for further calculations or interpretations. Here, understanding how to handle variables ensures the expression is easily manageable and closer to a basic form.

For instance, when dividing \( u \) and \( z \) in \( \frac{u}{u^3} \) and \( \frac{z^4}{z^8} \), we implement subtraction of exponents for faster and more efficient simplification:

• \( u \rightarrow u^{-2} \)
• \( z \rightarrow z^{-4} \)

This step changes the expression into a form that is much simpler and easier to understand. It also embodies the beauty of algebra: transforming what might seem complex at first glance into something elegant and straightforward.