- Multiply the numerators and denominators to form a single rational expression: \(\frac{7y^3(6x+5)}{105y}\).
- Simplify the numerical coefficients: \(\frac{7}{105} = \frac{1}{15}\).
- Simplify the variables by dividing \(\frac{y^3}{y} = y^2\).
- Distribute \(y^2\) to obtain the final result: \(\boxed{\frac{6xy^2+5y^2}{15}}\).
Explanation
- Setting up the multiplication
To multiply the given rational expressions, \(\left(\frac{7y^3}{5}\right)\) and \(\left(\frac{6x+5}{21y}\right)\), we first multiply the numerators and the denominators. The product is given by:
\[ \left(\frac{7y^3}{5}\right) \cdot \left(\frac{6x+5}{21y}\right) = \frac{7y^3(6x+5)}{5 \cdot 21y} \]
We assume \(y
eq 0\) to ensure the expression is defined.
- Simplifying the expression
Now we simplify the expression by dividing out common factors. Looking at the coefficients, we have a factor of \(7\) in the numerator and a \(21\) in the denominator. Since \(21 = 7 \cdot 3\), we can simplify:
\[ \frac{7}{21} = \frac{1}{3} \]
Next, we simplify the variable parts. We have \(y^3\) in the numerator and \(y\) in the denominator. Dividing \(y^3\) by \(y\) gives \(y^2\):
\[ \frac{y^3}{y} = y^2 \]
Putting these simplifications together:
\[ \frac{7y^3(6x+5)}{5 \cdot 21y} = \frac{1 \cdot y^2 \cdot (6x+5)}{5 \cdot 3} = \frac{y^2(6x+5)}{15} \]
Distributing \(y^2\) into the terms inside the parentheses:
\[ \frac{6xy^2 + 5y^2}{15} \]
This matches the fourth option provided.
Examples
This type of algebraic simplification is essential in fields like physics or engineering, where formulas involving multiple variables are combined. For instance, if you are calculating the total power output based on two different efficiency factors that involve different variables, you might end up with complex rational expressions. Simplifying them allows you to easily plug in values for your variables to find the final result, saving time and reducing the risk of calculation errors.