Problem 47 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ \frac{\left(x y^{2} z^{3}\right)^{4}}{\left(x^{3} y^{2} z\right)^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{y^2 z^9}{x^5} \).

Step 1: Apply the Power Rule

First, apply the power rule \( (a^m)^n = a^{mn} \) to both the numerator and the denominator. Numerator: \[ (x y^2 z^3)^4 = x^4 (y^2)^4 (z^3)^4 = x^4 y^8 z^{12} \]Denominator: \[ (x^3 y^2 z)^3 = (x^3)^3 (y^2)^3 (z)^3 = x^9 y^6 z^3 \].

Step 2: Simplify the Fraction

Now simplify the fraction by dividing the corresponding powers of like bases using the quotient rule \( \frac{a^m}{a^n} = a^{m-n} \). For \(x\): \[ \frac{x^4}{x^9} = x^{4-9} = x^{-5} \]For \(y\):\[ \frac{y^8}{y^6} = y^{8-6} = y^2 \]For \(z\):\[ \frac{z^{12}}{z^3} = z^{12-3} = z^9 \].

Step 3: Eliminate the Negative Exponent

Convert the expression with the negative exponent into positive exponents by rewriting them as fractions: \[ x^{-5} = \frac{1}{x^5} \].Thus, the expression becomes:\[ \frac{y^2 z^9}{x^5} \].

Key Concepts

Power Rule

The power rule is a powerful tool when simplifying expressions involving exponents. It states that when you raise a power to another power, you multiply the exponents. For example, under the power rule, \( (a^m)^n = a^{mn} \).
In algebra, this rule helps to break down complex expressions. When you multiply out the powers, you make the expression easier to handle.
In the given exercise, we applied the power rule separately to both the numerator and the denominator.

• Numerator: We expanded \( (xy^2z^3)^4 \) to become \( x^4 y^8 z^{12} \).
• Denominator: Similarly, \( (x^3 y^2 z)^3 \) expanded to \( x^9 y^6 z^3 \).

Applying the power rule efficiently manages and reduces complexity in expressions.

Quotient Rule

The quotient rule is central to algebraic simplification involving division of like bases. It works on the principle that when dividing exponential terms with the same base, you subtract the exponents.
This can be written as \( \frac{a^m}{a^n} = a^{m-n} \).
Applying the quotient rule is crucial for simplifying fractions with exponents and results in only one term per base.
Using the quotient rule in the exercise:

• x: We had \( \frac{x^4}{x^9} = x^{-5} \).
• y: We simplified \( \frac{y^8}{y^6} = y^2 \).
• z: For \( z \), it became \( \frac{z^{12}}{z^3} = z^9 \).

This step helps combine powers and eliminate redundancy, which further cleans up the expression.

Negative Exponents

Negative exponents can initially seem intimidating, but converting them is quite simple. They follow the rule \( a^{-n} = \frac{1}{a^n} \).
In essence, a negative exponent represents the reciprocal of the positive exponent.
In our exercise, we encountered \( x^{-5} \). Converting this involved turning the variable into the denominator:

• The expression \( x^{-5} \) becomes \( \frac{1}{x^5} \).

By handling negative exponents in this way, you ensure all exponents in the expression are positive, simplifying further calculations and interpretation.

Fraction Simplification

Fraction simplification is the final step in reducing expressions to their simplest form. After applying rules to manipulate exponents, simplifying fractions ensures the expression is easy to interpret and use.
In algebra, proper fraction simplification involves reducing both the numerator and the denominator by common factors and converting negative exponents.
In our example:

• We were left with \( \frac{y^2 z^9}{x^5} \) after handling all variations of powers and exponents.

Correctly simplified fractions reflect a clear and concise expression, avoiding unnecessary complexity, and ensuring clarity.