- The division of rational expressions follows the rule \(\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}\).
- Identify the dividend \(\frac{3 x^3 z^5}{5 y}\) and the divisor \(\frac{x^2 z^6}{20 y^3}\).
- Take the reciprocal of the divisor, which is \(\frac{20 y^3}{x^2 z^6}\).
- The correct first step is to multiply the dividend by this reciprocal: \(\boxed{\text{Take the reciprocal of the divisor to get } \frac{3 x^3 z^5}{5 y} \left(\frac{20 y^3}{x^2 z^6}\right)}\).
Explanation
- Understanding the Division Rule for Rational Expressions
To find the quotient of two rational expressions, we use the fundamental rule for division of fractions. Given two rational expressions \(\frac{P}{Q}\) and \(\frac{R}{S}\), the division \(\frac{P}{Q} \div \frac{R}{S}\) is equivalent to multiplying the dividend by the reciprocal of the divisor:
\[ \frac{P}{Q} \div \frac{R}{S} = \frac{P}{Q} \cdot \frac{S}{R} \]
In our specific case, the dividend is \(\frac{3 x^3 z^5}{5 y}\) and the divisor is \(\frac{x^2 z^6}{20 y^3}\). To perform this division, we need to identify the reciprocal of the divisor. The reciprocal of \(\frac{x^2 z^6}{20 y^3}\) is obtained by swapping its numerator and denominator, which yields \(\frac{20 y^3}{x^2 z^6}\).
- Identifying the Correct First Step
Now that we have identified the dividend and the reciprocal of the divisor, the first step in the division process is to set up the multiplication. Using the rule established in the previous step, we rewrite the division problem as:
\[ \frac{3 x^3 z^5}{5 y} \div \frac{x^2 z^6}{20 y^3} = \frac{3 x^3 z^5}{5 y} \cdot \frac{20 y^3}{x^2 z^6} \]
This matches the action of taking the reciprocal of the divisor and multiplying it by the dividend. Comparing this with the provided options, we can see that taking the reciprocal of the divisor to get \(\frac{3 x^3 z^5}{5 y} \left(\frac{20 y^3}{x^2 z^6}\right)\) is the correct first step.
Examples
In real-world applications, such as adjusting ingredient ratios in a professional recipe or scaling chemical concentrations in a laboratory, you often need to perform operations involving fractions and variables. For example, if you have a concentration of a substance represented as a ratio of chemicals per unit volume, dividing by another ratio requires multiplying by the reciprocal to correctly simplify the final concentration formula. Mastering this fundamental algebraic step ensures accuracy when scaling systems up or down in engineering, cooking, or scientific research.