Problem 90 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ \frac{3^{3} x^{4} y^{3} z}{3^{2} x y^{5} z^{5}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{3^{3} x^{4} y^{3} z}{3^{2} x y^{5} z^{5}}\) and rewrite it using only positive exponents. Answer: \(\frac{3x^3}{y^2 z^4}\)

Step 1: Identify the expression

The given expression is: $$ \frac{3^{3} x^{4} y^{3} z}{3^{2} x y^{5} z^{5}} $$

Step 2: Apply properties of exponents

Apply the quotient property of exponents for each variable, which states that \(\frac{a^m}{a^n} = a^{(m-n)}\). In our case, do this for the coefficients \((3^{3}/3^{2})\), and variables \(x\), \(y\), and \(z\). $$ \frac{3^3}{3^2} x^{(4-1)} y^{(3-5)} z^{(1-5)} \\ 3^{(3-2)} x^{3} y^{-2} z^{-4} $$

Step 3: Convert negative exponents to positive exponents

Using the rule \(a^{-n} = \frac{1}{a^n}\), we will turn the negative exponents for \(y\) and \(z\) into positive ones: $$ 3^{1} x^{3} \frac{1}{y^2} \frac{1}{z^4} \\ 3x^3\frac{1}{y^2 z^4} $$

Step 4: Write the final expression

Combine the terms to get the final expression using only positive exponents: $$ \frac{3x^3}{y^2 z^4} $$ The given expression simplified using only positive exponents is \(\frac{3x^3}{y^2 z^4}\).

Key Concepts

Exponent Rules

Exponent rules are foundational for simplifying algebraic expressions. They allow us to manipulate expressions involving powers of numbers and variables effortlessly. A key rule is the quotient of powers property, which states that \( \frac{a^m}{a^n} = a^{m-n} \). This explains how dividing two powers of the same base results in subtracting their exponents.
Another critical rule is the negative exponent rule. It tells us \(a^{-n} = \frac{1}{a^n}\). Thus, a negative exponent indicates the reciprocal of the base raised to the positive exponent. These rules are powerful tools for rewriting expressions with only positive exponents. Together, they form the backbone of simplifying complex exponent expressions.
Understanding and applying these rules helps transform mathematical expressions into simpler, more manageable forms.

Algebraic Expressions

Algebraic expressions consist of constants, variables, and exponents combined using arithmetic operations. They are the language of algebra and a means to represent mathematical relationships. In our expression \( \frac{3^3 x^4 y^3 z}{3^2 x y^5 z^5} \), we have a combination of coefficients and variable bases raised to different powers.

Breaking down such an expression involves identifying and aligning terms based on their bases. This allows us to apply the exponent rules correctly. Each of these parts—the coefficient \(3\), and the variable terms \(x\), \(y\), and \(z\)—must be handled separately but follow similar exponentiation principles.
Algebraic expressions allow for manipulation through well-defined rules, leading to simplifications and solutions to equations.

Simplifying Expressions

Simplifying expressions involves the application of mathematical rules to reduce them to their simplest form. This often means eliminating negative exponents or combining like terms. The goal is to rewrite the expression in a more compact and comprehensible way.

To simplify \( \frac{3^3 x^4 y^3 z}{3^2 x y^5 z^5} \) using positive exponents, we first used the quotient of powers rule to subtract the exponents of like bases. We simplified each base separately:

• Reduced \(3^3 / 3^2\) to \(3^1\).
• Simplified \(x^{4-1}\) to \(x^3\).
• Converted negative exponents \(y^{-2}\) and \(z^{-4}\) to \(\frac{1}{y^2}\) and \(\frac{1}{z^4}\).

This gives us a simpler expression: \( \frac{3x^3}{y^2 z^4} \), where all exponents are positive. By executing the correct algebraic operations, any expression can be systematically simplified to make solving equations easier or to better understand the relationship between terms.