Problem 26 - Technical Mathematics with Calculus

Question

Divide and reduce. Try some by calculator. $$\frac{5 a b c^{3}}{3 x^{2}} \div \frac{10 a c^{3}}{6 b x^{2}}$$

Step-by-Step Solution

Answer

\(b\)

Step 1: Write Out the Division as a Fraction

To divide two fractions, we convert the division problem into a multiplication problem by multiplying by the reciprocal of the second fraction. That is, we change \(\frac{5 a b c^{3}}{3 x^{2}} \div \frac{10 a c^{3}}{6 b x^{2}}\) into \(\frac{5 a b c^{3}}{3 x^{2}} \times \frac{6 b x^{2}}{10 a c^{3}}\).

Step 2: Multiply the Fractions

We multiply the numerators together and the denominators together: \(\frac{5 a b c^{3}}{3 x^{2}} \times \frac{6 b x^{2}}{10 a c^{3}} = \frac{5 a b c^{3} \times 6 b x^{2}}{3 x^{2} \times 10 a c^{3}}\).

Step 3: Cancel Common Factors

Reduce the fraction by canceling out common factors from the numerator and the denominator. In this case, \(a\), \(c^3\), \(x^2\), and \(b\) cancel out, and 5/10 reduces to 1/2, 6/3 reduces to 2/1: \(\frac{1 \times 2 \times b}{2 \times 1 \times a \times 1} = \frac{2b}{2}\).

Step 4: Simplify the Result

The fraction simplifies to \(\frac{2b}{2} = b\) as the 2 in the numerator and the denominator cancel each other out.

Key Concepts

Multiplying by the Reciprocal

When dividing fractions with variables, 'multiplying by the reciprocal' is a crucial step. Instead of direct division, we multiply the first fraction by the reciprocal of the second. The reciprocal of a fraction is created by swapping its numerator and denominator. For example, the reciprocal of \( \frac{10 a c^{3}}{6 b x^{2}} \) is \( \frac{6 b x^{2}}{10 a c^{3}} \).

When we have the division \( \frac{5 a b c^{3}}{3 x^{2}} \) divided by \( \frac{10 a c^{3}}{6 b x^{2}} \) it turns into multiplication: \( \frac{5 a b c^{3}}{3 x^{2}} \times \frac{6 b x^{2}}{10 a c^{3}} \). This maneuver allows us to avoid the more complicated division process, and instead use multiplication which most find simpler to handle, especially when variables are involved.

Reducing Algebraic Fractions

Reducing algebraic fractions involves simplifying expressions by removing common factors from the numerator and the denominator. It is akin to reducing numeric fractions to their lowest terms. Since we are dealing with variables and their exponents, we look to cancel out identical factors. With our given problem, we examine \( \frac{5 a b c^{3} \times 6 b x^{2}}{3 x^{2} \times 10 a c^{3}} \).

Looking at both the numerator and denominator, we identify the common factors such as \( a \) , \( b \) , \( c^{3} \) , and \( x^{2} \). These common factors can be divided out, meaning we 'reduce' the fraction, resulting in a much simpler expression.

Canceling Common Factors

Canceling common factors is directly connected to reducing algebraic fractions. This step is about efficiency and simplification. When the same variable or number appears in both the top and bottom of a fraction, they can be canceled out because anything divided by itself equals one. In our problem, after multiplying the numerators and denominators, we have both \( a \) and \( b \) appearing in both, allowing us to cancel them.

Moreover, the power of \( c^{3} \) also cancels out. Finally, the numerical factors \( 5/10 \) and \( 6/3 \) reduce to \( 1/2 \) and \( 2/1 \) respectively. Always remember, canceling is only valid if the factor appears in both the numerator and the denominator exactly in the same form or power.

Simplifying Algebraic Expressions

The last step is simplifying the algebraic expression to its most elementary form. Simplification might involve canceling common factors, combining like terms, or using algebraic identities. In the context of our fraction division problem, once we have canceled common factors, we are left with a simplified form, which then may still need further simplification.

After canceling, we may have numbers or variables that can still be simplified further. For instance, \( \frac{2b}{2} \) appears to have a common numerical factor of 2 in both numerator and denominator, which can be simplified to \( b \). This step is not just about reducing bulk; it's also about clarity and comprehension, making the resulting expressions cleaner and easier to work with in future mathematical procedures.