Problem 71 - College Algebra with Modeling and Visualization

Question

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \frac{5 r^{2} s t^{-3}}{25 r s^{-2} t^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{r \cdot s^3}{5 \cdot t^5} \).

Step 1: Simplify the Coefficient

First, look at the numerical coefficients in the expression: \( \frac{5}{25} \). Simplify \( \frac{5}{25} \) by reducing it to \( \frac{1}{5} \).

Step 2: Apply the Quotient Rule for Exponents to 'r'

Apply the quotient rule for exponents, which states \( \frac{a^m}{a^n} = a^{m - n} \), to the variable \( r \). Here, it becomes \( r^{2-1} = r^1 = r \).

Step 3: Apply the Quotient Rule for Exponents to 's'

Apply the quotient rule to \( s \). This results in \( s^{1 - (-2)} = s^{1 + 2} = s^3 \).

Step 4: Apply the Quotient Rule for Exponents to 't'

Apply the quotient rule to \( t \). This results in \( t^{-3 - 2} = t^{-5} \). Since we need positive exponents, \( t^{-5} \) is written as \( \frac{1}{t^5} \).

Step 5: Combine the Results

Combine the simplified components: The coefficient simplifies to \( \frac{1}{5} \), \( r \), and \( s^3 \), while \( t^{-5} \) becomes \( \frac{1}{t^5} \). The simplified expression is: \[ \frac{r \cdot s^3}{5 \cdot t^5} \].

Key Concepts

Quotient Rule

The Quotient Rule is an essential concept in simplifying expressions involving exponents. It helps us manage terms with the same base across a division. If you encounter expressions like \( \frac{a^m}{a^n} \), the Quotient Rule allows you to simplify it as \( a^{m-n} \). This means you subtract the exponent in the denominator from the exponent in the numerator.
This concept is particularly handy when you have terms like \( r^2 \) over \( r \), resulting in \( r^{2-1} = r^1 = r \).
The same rule applies to any similar terms within an expression. In the exercise, we use the Quotient Rule to simplify the variables \( r \), \( s \), and \( t \). By subtracting the exponents, you efficiently condense terms and bring simplicity to seemingly complex expressions.

Simplifying Expressions

Simplifying expressions is about making the expression cleaner and more manageable. You often start by dealing with coefficients and then follow with variables.

• First, handle any coefficients by reducing fractions to their simplest form as in this example where \( \frac{5}{25} \) reduces to \( \frac{1}{5} \).
• Apply the Quotient Rule to like terms, making sure to perform the required subtraction of exponents.
• Finally, multiply what remains to get a neat outcome.

By simplifying expressions, you make them easier to understand and work with, especially in larger problems where many terms and calculations could otherwise be overwhelming.

Positive Exponents

In mathematics, it's often preferable to express results using positive exponents. This keeps expressions clear and avoids negative powers which can seem unintuitive.
To convert a negative exponent to a positive one, you take the reciprocal of the term. For instance, \( t^{-5} \) becomes \( \frac{1}{t^5} \).

• This conversion makes calculations easier since positive exponents reinforce a straightforward multiplication process.
• It also aligns with many conventions in different applications of math, from algebra to calculus, providing consistency across mathematical work.
• In the final answer, formatting is key, arranging the terms so that the positive exponents are clearly presented.

Remember, positive exponents reflect a solid understanding of the manipulation of powers and show precision in mathematical communication.