Problem 20 - Algebra A Combined Function

Question

Find each quotient and simplify. See Examples 4 through 7. $$ \frac{7 a^{2} b}{3 a b^{2}} \div \frac{21 a^{2} b^{2}}{14 a b} $$

Step-by-Step Solution

Answer

The simplified quotient is \( \frac{14}{9b^2} \).

Step 1: Rewrite the Division as Multiplication

To divide by a fraction, you can multiply by its reciprocal. So, our problem becomes: \[ \frac{7a^2b}{3ab^2} \times \frac{14ab}{21a^2b^2} \]

Step 2: Multiply the Numerators

Multiply the numerators of each fraction: \[ 7a^2b \times 14ab = 98a^3b^2 \]

Step 3: Multiply the Denominators

Multiply the denominators of each fraction: \[ 3ab^2 \times 21a^2b^2 = 63a^3b^4 \]

Step 4: Combine the Results

Place the results from Steps 2 and 3 into a single fraction: \[ \frac{98a^3b^2}{63a^3b^4} \]

Step 5: Simplify the Fraction

Cancel out common terms in the numerator and denominator. - Simplify the coefficients: \( \frac{98}{63} = \frac{14}{9} \) - Simplify the powers of \(a\): \( \frac{a^3}{a^3} = 1 \)- Simplify the powers of \(b\): \( \frac{b^2}{b^4} = \frac{1}{b^2} \)After simplification, we get: \[ \frac{14}{9b^2} \]

Step 6: Final Simplified Quotient

The final simplified form of the quotient is: \[ \frac{14}{9b^2} \]

Key Concepts

Quotient

In mathematics, a quotient is the result you get when you divide one number or quantity by another. It's essential to understand quotients, especially when dealing with algebraic fractions.
Algebraic expressions often lead to complex quotients where variables and coefficients are involved. The key is to look at both the numerator and the denominator separately and use properties of division to simplify the quotient as much as possible.
Remember, when you see the fraction bar, it signifies division. So in algebra, just like numbers, you can divide expressions to find a quotient. This is valuable in simplifying expressions and solving equations.

Simplification

Simplification in algebra involves reducing a complex expression to its simplest form. When you simplify a quotient of algebraic fractions, you make it easier to understand by eliminating common factors.
The process involves:

• Canceling out terms that appear in both the numerator and the denominator.
• Reducing coefficients to their smallest integer values.
• Simplifying the exponents on variables by subtracting powers if they appear in both parts of the fraction.

Simplifying helps in making an expression clearer and more manageable. It also ensures accuracy for any further calculations or solutions required.

Reciprocal

A reciprocal involves inverting a fraction. Essentially, you swap the numerator and denominator to form the reciprocal.
This concept is crucial in dividing fractions. Instead of dividing, you multiply by the reciprocal, an essential step when handling complex fractions.
For example, the reciprocal of \( \frac{21a^2b^2}{14ab} \) is \( \frac{14ab}{21a^2b^2} \). This switch makes multiplication straightforward. Recognizing and applying reciprocals is a handy tool in algebra, simplifying the division of fractions into manageable multiplication tasks.

Fraction Multiplication

Multiplying fractions is more straightforward than it might appear. You multiply the numerators together and the denominators together separately. For example, with the fractions \( \frac{7a^2b}{3ab^2} \) and \( \frac{14ab}{21a^2b^2} \), you do the simple operation:

• The numerators: \( 7a^2b \times 14ab = 98a^3b^2 \)
• The denominators: \( 3ab^2 \times 21a^2b^2 = 63a^3b^4 \)

Once multiplied, you form a new fraction, and from this point, simplification follows. Understanding multiplication with fractions helps perform calculations that seem daunting, making algebra less intimidating.