Problem 84 - Elementary Algebra

Question

Write \(\frac{x^{3} y^{-5}}{z^{-4}}\) so that only positive exponents appear.

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\frac{x^{3} y^{-5}}{z^{-4}}\) with only positive exponents. Answer: \(\frac{x^{3}}{z^4 y^5}\)

Step 1: Rewrite using the negative exponent rule

Recall that \(a^{-n} = \frac{1}{a^n}\). Use this rule to rewrite the negative exponents in the given expression: \(\frac{x^{3} y^{-5}}{z^{-4}} = \frac{x^{3}}{z^4} \times \frac{1}{y^5}\). Notice that now we have two fractions that can be combined.

Step 2: Combine the fractions

Multiply the two fractions from step 1: \(\frac{x^{3}}{z^4} \times \frac{1}{y^5} = \frac{x^{3} \cdot 1}{z^4 \cdot y^5}\)

Step 3: Simplify the expression

Our final expression is \(\frac{x^{3}}{z^4 y^5}\). All the exponents are positive and the expression is simplified.

Key Concepts

Negative Exponents

Negative exponents can seem tricky initially, but they are intuitive once you understand how they work. When we have an expression like \(a^{-n}\), it means that instead of multiplying, we are actually dividing by \(a\) raised to the \(n\)th power. In simpler terms, a negative exponent indicates the reciprocal of the base raised to the positive version of the exponent.

For instance, \(a^{-2}\) can be rewritten as \(\frac{1}{a^2}\). This concept allows us to transform expressions with negative exponents into those with only positive exponents, making computation easier and less error-prone.

In our example, we started with \(y^{-5}\) and \(z^{-4}\) inside the fraction. By rewriting these terms, \(y^{-5}\) became \(\frac{1}{y^5}\), and \(z^{-4}\) turned into \(z^4\) in the denominator, ultimately helping us simplify the expression to its positive-exponent form.

Fraction Simplification

Simplifying fractions is a fundamental skill in algebra. When you're handling fractions, especially with exponents, it can be helpful to make the expression as manageable as possible.

In our exercise, we worked with the expression \(\frac{x^{3}}{z^{-4}} \times \frac{1}{y^5}\). This might seem complex at first, but by multiplting these fractions, we consolidated the terms into a single fraction.

• Numerator: combine all variables and coefficients in the top part.
• Denominator: do the same for the bottom part, but ensure all exponents are positive.

This resulted in \(\frac{x^{3}}{z^4 y^5}\), a single clean fraction with fully simplified and positive exponents. Simplified fractions are easier to work with and interpret.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. When working with them, especially with exponents involved, it's crucial to apply rules correctly at each step.

In our original problem, \(\frac{x^{3} y^{-5}}{z^{-4}}\), we aim to simplify and have only positive exponents appear. This requires careful application of exponent rules (like turning negatives into reciprocal form) and proper fraction sense.

These expressions are fundamental building blocks in algebra, used to solve equations, model real-world situations, or represent unknown quantities. Understanding how to manipulate and simplify them allows for clearer solutions and prepares you for more advanced mathematical concepts involving equations and inequalities.