Problem 73 - Algebra 2 and Trigonometry

Question

In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{20 x^{0} y^{-5}}{4 x^{-1} y^{5}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 5x \cdot \frac{1}{y^{10}} \).

Step 1: Simplify the Coefficient

First, simplify the numerical coefficients of the fraction. Here, the coefficient is \( \frac{20}{4} \). This simplifies to \( 5 \).

Step 2: Apply the Exponent Rule for Division

Use the exponent rule \( \frac{a^m}{a^n} = a^{m-n} \) to simplify the powers of \( x \) and \( y \):\[ \frac{x^0}{x^{-1}} = x^{0+1} = x^1 = x \].

Step 3: Simplify the Exponents of y

Similarly, for \( y \),\[ \frac{y^{-5}}{y^5} = y^{-5-5} = y^{-10} \].

Step 4: Combine the Simplified Parts

Combine the results of Steps 1, 2, and 3: \( 5 \cdot x \cdot y^{-10} \).

Step 5: Remove Negative Exponent by Using the Inverse

Transform the negative exponent \( y^{-10} \) by taking its reciprocal: \[ y^{-10} = \frac{1}{y^{10}} \]. Write the expression as a product: \( 5x \cdot \frac{1}{y^{10}} \).

Key Concepts

Product Without Denominator

When we encounter fractions or quotients in algebra, converting them into a "product without a denominator" can often simplify our calculations. The main goal is to express all parts of a quotient as a multiplication, which is typically easier to handle and more straightforward for further algebraic manipulation. A "product without denominator" entails rewriting the expression such that any division between terms is represented through multiplication instead. This process involves understanding and applying rules of exponents efficiently.

For example, consider the expression:

• \( \frac{20 x^{0} y^{-5}}{4 x^{-1} y^{5}} \)

First, simplify the numerical coefficients: \( \frac{20}{4} = 5 \). Then, address the variables using exponent rules, which will help eliminate the denominator. By exploiting the property \( \frac{a^m}{a^n} = a^{m-n} \), we arrange each variable as a standalone term in a product relation. This transformation allows the expression to be written consistently as products, offering clarity and ease of further manipulation.

Exponent Rules

Exponent rules are crucial for algebraic simplification. They provide the framework to manipulate expressions involving powers, making calculations more comprehensible. Let's delve into the two main rules applied in this context:

• Division Rule: This states that \( \frac{a^m}{a^n} = a^{m-n} \). When exponents of the same base are divided, the result is simply a subtraction of the exponents.
• Zero Exponent Rule: Any number (except zero) raised to the power of zero equals one: \( a^0 = 1 \). This implies \( x^0 \) becomes 1, simplifying any term drastically.

In the given expression \( \frac{x^0}{x^{-1}} = x^{0+1} = x \), we utilize these rules effectively:

• \( x^0 = 1 \), simplifying the unneeded variable
• Add the exponent of \(-1\) from the denominator to the numerator due to the division rule

Exponent rules such as these streamline complex expressions, producing organized and concise results.

Negative Exponents

Understanding negative exponents is essential for proper algebraic manipulation. A negative exponent indicates reciprocal action, transforming powers to reciprocals. To illustrate, \( a^{-n} \) equates to \( \frac{1}{a^n} \). This property allows negative powers to shift between numerator and denominator depending on the expression's needs.Consider the term \( y^{-10} \):

• Instead of writing \( y^{-10} \), you express it as \( \frac{1}{y^{10}} \).

This approach aligns with transforming divisions into multiplications, creating a product without a fraction. For learners, perceiving negative exponents as instructions to "flip" part of a fraction solidifies comprehension and simplifies their application.

Thus, understanding negative exponents grants the ability to reconfigure complex expressions into their simplest forms, ready for evaluation or further manipulation.