Problem 21 - A Graphical Approach to College Algebra

Question

Find each product or quotient. $$\frac{15 p^{3}}{9 p^{2}} \div \frac{6 p}{10 p^{2}}$$

Step-by-Step Solution

Answer

The final result is \( \frac{25p^2}{9} \).

Step 1: Write the Division as a Multiplication

To divide by a fraction, multiply by its reciprocal. So, rewrite the division problem as multiplication of fractions:\[ \frac{15p^3}{9p^2} \times \frac{10p^2}{6p} \]

Step 2: Simplify the First Fraction

Simplify the fraction \( \frac{15p^3}{9p^2} \). Divide the coefficients (15 and 9) and subtract the exponents of \( p \):\[ \frac{15}{9} = \frac{5}{3} \] \[ p^{3-2} = p^1 = p \] So, \( \frac{15p^3}{9p^2} = \frac{5p}{3} \).

Step 3: Simplify the Second Fraction

Next, simplify \( \frac{10p^2}{6p} \). Divide the coefficients (10 and 6) and subtract the exponents of \( p \):\[ \frac{10}{6} = \frac{5}{3} \] \[ p^{2-1} = p^1 = p \] So, \( \frac{10p^2}{6p} = \frac{5p}{3} \).

Step 4: Multiply the Simplified Fractions

Now multiply the simplified fractions from Steps 2 and 3:\[ \frac{5p}{3} \times \frac{5p}{3} \] Multiply the numerators and the denominators separately:\[ \frac{5p \times 5p}{3 \times 3} = \frac{25p^2}{9} \]

Step 5: Final Result

There are no further simplifications possible, so the product of the original expression is\[ \frac{25p^2}{9} \].

Key Concepts

Simplification of Fractions

Simplification of fractions involves reducing the fraction to its simplest form. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

For instance, in simplifying \( \frac{15p^3}{9p^2} \), the GCD of 15 and 9 is 3. Dividing both by 3, we get \( \frac{5}{3} \). Also, when powers of the same base are divided, the exponents are subtracted: \( p^{3-2} = p^1 = p \). Thus, \( \frac{15p^3}{9p^2} \) simplifies to \( \frac{5p}{3} \). Make sure that you always simplify fractions before performing further operations.

Multiplying and Dividing Fractions

When dealing with fractions, division can be turned into multiplication. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
For instance, changing the division \( \frac{15p^3}{9p^2} \div \frac{6p}{10p^2} \) into multiplication involves taking the reciprocal of the second fraction:
\[ \frac{15p^3}{9p^2} \times \frac{10p^2}{6p} \]When multiplying fractions:

• Multiply the numerators together.
• Multiply the denominators together.

Always simplify the fractions first to make calculations easier and the result in its simplest form. Once simplified, as in the example above, the process becomes straightforward and manageable.

Exponents in Algebra

Exponents are a way to express repeated multiplication of the same base. In algebra, handling exponents efficiently is crucial, especially when simplifying terms.

• When multiplying like bases, add the exponents. For example, \( p^a \times p^b = p^{a+b} \).
• When dividing like bases, subtract the exponents. For example, \( p^a \div p^b = p^{a-b} \).

In our example, the simplification process involves subtracting exponents when dividing fractions: \( p^{3-2} = p \). Furthermore, when multiplying simplified results such as \( \frac{5p}{3} \times \frac{5p}{3} \), the exponents of \( p \) are added to give \( p^2 \). Understanding these rules helps make complex algebraic expressions more manageable. Be attentive to exponent rules as they form the cornerstone for algebraic manipulation.