Problem 47 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Question: Rewrite the expression \(5 x^{2} y^{2} z^{-5}\) using only positive exponents. Answer: \(\frac{5 x^{2} y^{2}}{z^{5}}\)

Step 1: Rewrite the given expression using the inverse property of exponents

$$5 x^{2} y^{2} z^{-5}$$

Step 2: Apply the rule \(a^{-n}=\frac{1}{a^{n}}\)

$$5 x^{2} y^{2} \cdot \frac{1}{z^{5}}$$

Step 3: Simplify the expression

$$\frac{5 x^{2} y^{2}}{z^{5}}$$

Key Concepts

Inverse Property of Exponents

When dealing with exponents in algebra, it's crucial to understand how negative exponents are handled. The inverse property of exponents is a simple yet fundamental concept that helps us rewrite expressions with negative exponents into a format that only uses positive exponents. This is particularly handy when we want to simplify and solve algebraic problems.

The rule is straightforward: for any nonzero number 'a' and a positive integer 'n', the expression with a negative exponent can be rewritten as the reciprocal of the base raised to the positive exponent. In mathematical terms: \( a^{-n} = \frac{1}{a^n} \). This trivializes dealing with negative exponents because you can easily flip them into a fraction with a positive exponent in the denominator.

Let's apply this rule to the exercise \(5 x^{2} y^{2} z^{-5}\). The variable \(z\) has a negative exponent, which means we can apply the inverse property. By doing so, we transform \(z^{-5}\) into \(\frac{1}{z^{5}}\), providing us with a positive exponent and a more straightforward expression to work with, \(\frac{5 x^{2} y^{2}}{z^{5}}\).

Simplifying Expressions

Once we've handled any negative exponents, the next step is to simplify expressions. Simplification is all about making complex algebraic expressions easier to understand and work with. It often involves combining like terms, reducing fractions, and eliminating any factors common to the numerator and denominator. If you have ever cooked a meal and tried to reduce the sauce to get the richest flavor, then you’ve simplified in real life! It’s about distilling algebra to its most potent form.

In our example, after applying the inverse property of exponents, we're left with the expression \(\frac{5 x^{2} y^{2}}{z^{5}}\). This expression is already relatively simplified; there are no like terms to combine, and it's clear what the expression represents. However, in more complicated cases, simplification might involve additional steps such as factoring polynomials or canceling terms.

We should always aim to present our final answer in the simplest form possible to demonstrate a clear understanding of the mathematical concepts involved and to communicate our results effectively.

Algebraic Manipulation

Algebraic manipulation is an umbrella term that refers to the various methods used to transform expressions and equations. This could involve employing properties of exponents like the inverse property, expanding products, factoring, and even more complex techniques. Mastery in algebraic manipulation reflects a deep understanding of algebra's language and its rules.

In the solution to our exercise, this concept is showcased when we altered \(z^{-5}\) using the inverse property of exponents. This step is a form of algebraic manipulation—altering the structure without changing the value. An algebraic magician comprehensively understands when to apply certain maneuvers to simplify an expression or resolve an equation effectively.

Further manipulation of \(\frac{5 x^{2} y^{2}}{z^{5}}\) might not be necessary in this case, but for multifaceted expressions or equations, knowing how to wield these tools skillfully can be the key to unlocking even the most perplexing algebraic puzzles.