Problem 69 - 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. $$ \frac{6 a^{2} b^{-3}}{4 a b^{-2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3a}{2b} \).

Step 1: Simplify the Coefficients

The given expression is \( \frac{6 a^{2} b^{-3}}{4 a b^{-2}} \). Start by simplifying the coefficients (numbers) 6 and 4 by dividing both by their greatest common divisor, which is 2. This gives us \( \frac{3}{2} \).

Step 2: Apply the Quotient Rule for the Exponents of 'a'

Using the quotient rule \( \frac{x^m}{x^n} = x^{m-n} \), simplify the \( a \) terms: \( \frac{a^{2}}{a^{1}} = a^{2-1} = a^{1} \). So the expression becomes \( \frac{3 a}{2 b^{-3}}{b^{-2}} \).

Step 3: Apply the Quotient Rule for the Exponents of 'b'

Similarly, apply the quotient rule to the \( b \) terms: \( \frac{b^{-3}}{b^{-2}} = b^{-3+2} = b^{-1} \). Now the expression is \( \frac{3a}{2} b^{-1} \).

Step 4: Convert Negative Exponent to Positive Exponent for 'b'

A negative exponent means division, so \( b^{-1} \) is the same as \( \frac{1}{b} \). Therefore, the whole expression becomes \( \frac{3a}{2b} \).

Key Concepts

Simplifying Expressions

Simplifying expressions is about making a mathematical expression easier to understand and work with. This usually involves reducing the complexity of the expression while maintaining its original value. For the expression \( \frac{6 a^{2} b^{-3}}{4 a b^{-2}} \), simplification starts with dividing the coefficients and applying the rules for exponents.

• Identify the coefficients: These are the numbers in front of variables, which in our case are 6 and 4.
• Find the greatest common divisor (GCD) of these coefficients, which is 2, and divide. This gives us \( \frac{3}{2} \).
• Simplify the variables by using exponent rules.

After simplifying the coefficients, our focus shifts to applying exponent rules to simplify the variables.

Quotient Rule

The quotient rule for exponents is a powerful tool when handling division in algebraic expressions. Essentially, it states that when you divide like bases, you subtract their exponents: \( \frac{x^m}{x^n} = x^{m-n} \).

• Apply this to the \( a \) terms: \( \frac{a^{2}}{a^{1}} = a^{2-1} = a^{1} \) or simply \( a \).
• Next, apply this to the \( b \) terms: \( \frac{b^{-3}}{b^{-2}} = b^{-3+2} = b^{-1} \).

The quotient rule simplifies our expression to: \( \frac{3a}{2} b^{-1} \), combining the simplified coefficients and variables.

Negative Exponents

Negative exponents can be tricky at first, but the key is remembering they imply division by that base raised to the positive of the exponent. For any non-zero number \( x \), \( x^{-n} = \frac{1}{x^n} \).

• In our expression, \( b^{-1} \) translates to \( \frac{1}{b} \).
• This helps in repositioning parts of an expression, making it either part of a numerator or denominator.

By recognizing that \( b^{-1} \) can be simplified to \( \frac{1}{b} \), we can further simplify our expression.

Positive Exponents

Using positive exponents in your final answer is usually the standard for simplified expressions. Positive exponents are straightforward as they reflect numbers being multiplied a certain number of times.

• In our solution, after handling negative exponents, we write the expression with positive exponents whenever possible.
• This ensures clarity and follows the convention that expressions without negative exponents are more simplified.

Therefore, the idea is to convert any negative exponents, like \( b^{-1} \), to positive by adjusting the position in a fraction. Our expression \( \frac{3a}{2b} \) is its simplest form using only positive exponents.