Problem 11 - Elementary Algebra

Question

When simplifying the terms for the following problems, write each so that only positive exponents appear. $$ \frac{5 x^{3} y^{-2}}{z^{-4}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{5x^3y^{-2}}{z^{-4}}\) so that it only has positive exponents. Answer: \(5x^3z^4\frac{1}{y^2}\)

Step 1: Identify the terms with negative exponents.

In the given expression, notice that there are two terms with negative exponents: \(y^{-2}\) in the numerator and \(z^{-4}\) in the denominator.

Step 2: Apply the negative exponent rule for \(y^{-2}\).

Recall that the negative exponent rule states that \(a^{-n}=\frac{1}{a^n}\). Therefore, \(y^{-2}=\frac{1}{y^2}\). We can now rewrite the given expression as: $$ \frac{5 x^3 \frac{1}{y^2}}{z^{-4}} $$

Step 3: Apply the negative exponent rule for \(z^{-4}\).

Using the same rule, we get \(z^{-4}=\frac{1}{z^4}\). Now, we can rewrite the expression again as: $$ \frac{5 x^3 \frac{1}{y^2}}{\frac{1}{z^4}} $$

Step 4: Combine the fractions

Now that we have only positive exponents, we can combine the fractions. To do this, we will multiply the numerator with the reciprocal of the denominator: $$ \frac{5 x^3 \frac{1}{y^2}}{\frac{1}{z^4}} = 5 x^3 \frac{1}{y^2} \cdot z^4 $$

Step 5: Simplify the expression

Finally, we can simplify the expression by combining the terms: $$ 5 x^3 \frac{1}{y^2} \cdot z^4 = 5 x^3 z^4 \cdot \frac{1}{y^2} $$ So the simplified expression with only positive exponents is: $$ 5x^3z^4\frac{1}{y^2} $$

Key Concepts

Negative Exponents

Negative exponents are a way to express division using exponents, making it easier to manipulate mathematical expressions. If you see a term like \(y^{-2}\), it means you write it as \(\frac{1}{y^2}\). This transformation is known as the negative exponent rule, and it's a key concept in algebra. By rewriting terms with negative exponents as fractions with positive exponents, you can help simplify complex expressions. This rule holds for any base: \(a^{-n} = \frac{1}{a^n}\). Understanding and applying this rule allows you to turn challenging expressions into simpler, more manageable forms.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their simplest form. This often involves combining like terms, applying algebraic rules, and rewriting expressions so that they're easier to work with. Start by identifying terms that can be rewritten using simplification rules, such as those involving exponents.
In our exercise, negative exponents were simplified to make the expression simpler. This included rewriting \(y^{-2}\) as \(\frac{1}{y^2}\), and \(z^{-4}\) as \(\frac{1}{z^4}\). Once rewritten, these terms were multiplied appropriately to produce a clean expression with only positive exponents.
Simplifying isn't just about making expressions look nicer; it's about making math problems less tricky to solve. Often, a simplified expression will make further calculations more straightforward, enabling you to solve mathematical problems more efficiently.

• Look for negative exponents to convert them.
• Apply arithmetic and algebraic rules.
• Combine all parts to create a single simplified expression.

Algebraic Fractions

Algebraic fractions combine the principles of fractions with algebraic expressions. When working with these, you'll encounter all the rules of fractions — like finding common denominators — but with the added layer of variables and exponents. In our example, the initial expression contained algebraic fractions from converting negative exponents.
Simplifying algebraic fractions usually involves:

• Converting negative exponents to positive by rewriting them as fractions.
• Working with multiplication and division of fractions.

To simplify \( \frac{5 x^{3} y^{-2}}{z^{-4}} \), negative exponents were converted, turning the expression into a multiplication problem: \( 5 x^3 \frac{1}{y^2} \cdot z^4 \). The task is then to combine these into a simplified form.
The goal is to leave terms that are easy to combine and calculate, expressed entirely with positive exponents. This provides a cleaner and more practical expression for further mathematical operations.