Problem 22 - A Graphical Approach to College Algebra

Question

Find each product or quotient. $$\frac{3 r^{2}}{9 r^{3}} \div \frac{8 r^{3}}{6 r}$$

Step-by-Step Solution

Answer

\( \frac{1}{4r^{3}} \)

Step 1: Rewrite the Expression

Start with the given expression: \( \frac{3 r^{2}}{9 r^{3}} \div \, \frac{8 r^{3}}{6 r} \). To simplify this, we rewrite the division as a multiplication by the reciprocal. This gives us: \( \frac{3 r^{2}}{9 r^{3}} \times \, \frac{6 r}{8 r^{3}} \).

Step 2: Simplify the First Fraction

Simplify \( \frac{3 r^{2}}{9 r^{3}} \) by canceling common factors. The numbers 3 and 9 share a factor of 3: \( \frac{3}{9} = \frac{1}{3} \). For the powers of \( r \), subtract the exponents: \( r^{2 - 3} = r^{-1} \). So we have: \( \frac{1}{3r} \).

Step 3: Simplify the Second Fraction

Simplify \( \frac{6 r}{8 r^{3}} \) by canceling common factors. The numbers 6 and 8 share a factor of 2: \( \frac{6}{8} = \frac{3}{4} \). For the powers of \( r \), subtract the exponents: \( r^{1 - 3} = r^{-2} \). So we have: \( \frac{3}{4r^{2}} \).

Step 4: Multiply the Fractions

Now, multiply the reduced fractions: \( \frac{1}{3r} \times \frac{3}{4r^{2}} \). The numerators multiply to \( 1 \times 3 = 3 \). The denominators multiply to \( 3r \times 4r^{2} = 12r^{3} \), giving us the product: \( \frac{3}{12r^{3}} \).

Step 5: Simplify the Product

Simplify \( \frac{3}{12r^{3}} \) by dividing both the numerator and the denominator by 3. The result is \( \frac{1}{4r^{3}} \).

Key Concepts

Simplifying Fractions

Fractions are simplified by reducing them to their simplest form. This makes calculations easier and results clearer. In a fraction like \( \frac{3}{9} \), both the numerator (3) and the denominator (9) have a common factor, which is 3.
Simplifying involves dividing both the numerator and the denominator by their greatest common factor (GCF).

• For \( \frac{3}{9} \), divide both by 3 to get \( \frac{1}{3} \).

When variables with exponents appear in fractions, simplify by subtracting the exponents of like bases when dividing.

• For \( r^2 \) and \( r^3 \), the expression simplifies to \( r^{2-3} = r^{-1} \).

This method is important for ensuring our fractions are easy to work with in further calculations.

Exponents and Powers

Exponents denote how many times a number, known as the base, is multiplied by itself. An expression like \( r^3 \) means \( r \times r \times r \). Understanding exponents is crucial in algebra for simplifying expressions, especially with powers involved.
To simplify expressions with exponents, use laws of exponents, such as the subtraction rule for division:

• When dividing like bases, subtract the exponents \( a^{m} / a^{n} = a^{m-n} \). So \( r^{3}/r^{2} = r^{3-2} = r \).
• Remember that any number to the power of 0 is 1: \( a^0 = 1 \).

This helps make calculations accurate and manageable when losing track of powers in a complex problem.

Division of Fractions

Dividing fractions involves multiplying the first fraction by the reciprocal of the second. This might seem odd at first, but it's a surprisingly straightforward method. The reciprocal of a fraction is simply obtained by swapping the numerator and the denominator.
To divide fractions:

• Change the division to multiplication \( \frac{a}{b} \div \frac{c}{d} \to \frac{a}{b} \times \frac{d}{c} \).
• Multiply across the numerators and denominators \( \frac{a \times d}{b \times c} \).

For example, dividing \( \frac{3r^2}{9r^3} \div \frac{8r^3}{6r} \) is handled by flipping the second fraction and converting the division to multiplication. Then continue by simplifying across the numerators and denominators. Mastering this transforms complex fractional expressions into simpler results, which are easier to interpret and use.