Problem 17 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \frac{-5 x^{3} y^{3} z^{4}}{20 x^{3} y^{7} z^{4}} $$

Step-by-Step Solution

Answer

\( \frac{-1}{4y^{4}} \)

Step 1: Factor the Constants

First, look at the coefficients in the fraction. You have -5 in the numerator and 20 in the denominator. Simplify \( \frac{-5}{20} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This results in \( \frac{-1}{4} \).

Step 2: Simplify the x-terms

Both the numerator and the denominator have a term involving \( x^{3}\). Dividing \( x^{3} \) by \( x^{3} \) results in \( x^{0} = 1 \). Thus, the \( x \)-terms cancel out completely.

Step 3: Simplify the y-terms

For the \( y \)-terms, apply the quotient rule for exponents: \( \frac{y^{3}}{y^{7}} = y^{3-7} = y^{-4} \). This simplifies to \( \frac{1}{y^{4}} \).

Step 4: Simplify the z-terms

Finally, for the \( z \)-terms, apply the quotient rule: \( \frac{z^{4}}{z^{4}} = z^{0} = 1 \). Hence, the \( z \)-terms cancel out completely.

Step 5: Combine Simplified Results

After simplifying each part, combine the results to create the simplified expression: \( \frac{-1}{4y^{4}} \).

Key Concepts

Quotient Rule for Exponents

When dealing with algebraic fractions that include variables with exponents, the quotient rule for exponents is incredibly useful. This rule tells us how to simplify expressions when you're dividing exponential terms with the same base. It's simple: subtract the exponent in the denominator from the exponent in the numerator. For example, the rule for any base 'a' is expressed mathematically as follows: \[\frac{a^m}{a^n} = a^{m-n}\]This means if you have \( y^3 \) in the numerator and \( y^7 \) in the denominator, you subtract the exponents: \( 3-7 \), which results in \( y^{-4} \). This can be further simplified: because the exponent is negative, the term moves to the denominator, giving you \( \frac{1}{y^4} \). In this way, the quotient rule simplifies and reduces complex expressions quickly.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept in simplifying fractions, especially when dealing with coefficients. The GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder. By finding the GCD, you can reduce fractions to their simplest form.

• Identify the coefficients: In the expression \( \frac{-5}{20} \), both -5 and 20 are the numbers involved.
• Find their GCD: The GCD of 5 and 20 is 5, as 5 is the largest number that divides both numbers evenly.
• Simplify by dividing each number by the GCD: \( \frac{-5}{5} = -1 \) and \( \frac{20}{5} = 4 \), resulting in \( \frac{-1}{4} \).

Simplifying the coefficients helps in minimizing the numerical complexity of the fractions you work with.

Canceling Variables

Canceling variables is a straightforward method in simplifying algebraic fractions. It operates similarly to simplifying numeric fractions, where common factors in both numerator and denominator are eliminated. This process effectively "removes" terms if they equal 1.When we say we cancel terms, such as \( x^3 \) in both the numerator and denominator, we essentially perform this operation:

• Divide \( x^3 \) by \( x^3 \), which mathematically results in \( x^0 \).
• Since \( x^0 \) equals 1, this means \( x^3 \) cancels out entirely.

The same principle applies for other terms with the same base and exponent in both the numerator and denominator, like \( z^4 \). By canceling these variables, we simplify the expression without altering its value.