Problem 72 - 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. $$ \frac{36 r^{-1}(s t)^{2}}{9(r s)^{2} t^{-1}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{4t^3}{r^3}\).

Step 1: Simplify the Fraction

We start by simplifying the numerical portion of the fraction \(\frac{36}{9}\), which equals 4. Our expression becomes \[4 \times \frac{r^{-1}(s t)^2}{(r s)^{2} t^{-1}}.\]

Step 2: Apply Power to a Power Rule

Rewrite \((s t)^{2}\) as \(s^2 t^2\), and \((r s)^{2}\) as \(r^2 s^2\). This gives us the new expression: \[4 \times \frac{r^{-1} s^2 t^2}{r^2 s^2 t^{-1}}.\]

Step 3: Simplify the Exponents

Simplify the expression by applying the rules of exponents for division: subtract the exponents across the fraction. This gives us: \[4 \times r^{-1-2} s^{2-2} t^{2+1} = 4 \times r^{-3} s^{0} t^{3}.\]

Step 4: Simplify Using Positive Exponents

Since \(s^0 = 1\), we have \(4 \times r^{-3} \times t^3\). Rewrite \( r^{-3} \) as \( \frac{1}{r^3}\) to obtain only positive exponents: \[4 \times \frac{t^3}{r^3} = \frac{4t^3}{r^3}.\]

Key Concepts

Rules of Exponents

Exponents are powerful tools in algebra that help us manage and simplify expressions involving powers. Several key rules define how we can manipulate exponents effectively.

• Product of Powers Rule: When multiplying two terms with the same base, add their exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).
• Quotient of Powers Rule: When dividing terms with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Zero Exponent Rule: Any non-zero base raised to the power of zero is 1: \( a^0 = 1 \).
• Negative Exponent Rule: A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent: \( a^{-n} = \frac{1}{a^n} \).

Applying these rules allows us to simplify expressions efficiently. In our original exercise, the application of these exponent rules helps simplify the complex fraction to an expression with only positive exponents.

Fraction Simplification

Simplifying fractions involves reducing the fraction to its simplest form. Here, we use the rules of exponents to operate on terms separately and simplify them.
Begin by simplifying the numerical coefficients. For example, the fraction \( \frac{36}{9} \) simplifies to 4.
Next, apply the exponent rules. Simplify each component with the same base by subtracting exponents in the case of a division, like with \( \frac{r^{-1}}{r^2} \) giving \( r^{-1-2} = r^{-3} \).
Another example is \( \frac{(s t)^2}{(r s)^2 t^{-1}} \) where each bracketed term is simplified using the appropriate exponent rules. \( (s t)^2 \) becomes \( s^2 t^2 \) and \( (r s)^2 \) becomes \( r^2 s^2 \).
By reducing the expression step by step, you find it easier to handle and eventually arrive at a simple result.

Positive Exponents

Positive exponents indicate that numbers are multiplied by themselves a certain number of times, which is more intuitive in understanding and simplifying mathematical expressions.
Negative exponents, on the other hand, imply division. By rewriting them as positive exponents, we can streamline calculations and the expressions' interpretation.
In our exercise, the simplification process involved converting negative exponents into positive ones. For example, \( r^{-3} \) is rewritten as \( \frac{1}{r^3} \). Having positive exponents is preferable as they are easier to read and manipulate.
Thus, we reached the final expression \( \frac{4t^3}{r^3} \), which is visually simpler and easier to interpret. Working with positive exponents makes understanding and verifying mathematical operations more straightforward.