Simplify ((4by)/(3y^3))÷((2b)/(9y))
The problem requires you to simplify a complex fraction expression. Specifically, you are asked to divide one rational expression, \(\frac{4by}{3y^3}\), by another rational expression, \(\frac{2b}{9y}\). This involves manipulating the numerators and denominators of both expressions to find a simplified form. In the process, you would likely factor out common terms and apply the rule that dividing by a fraction is equivalent to multiplying by its reciprocal.
$\frac{4 b y}{3 y^{3}} \div \frac{2 b}{9 y}$
Answer
Solution:
Step 1
To simplify the division of two fractions, multiply the first fraction by the reciprocal of the second fraction. Thus, we have $\frac{4by}{3y^3} \times \frac{9y}{2b}$.
Step 2
Combine the numerators and denominators from both fractions. This results in $\frac{4by \cdot 9y}{3y^3 \cdot 2b}$.
Step 3
Simplify by combining like terms.
Step 3.1
Combine $y$ terms by adding their exponents. We get $\frac{4b(y \cdot y) \cdot 9}{3y^3 \cdot 2b}$.
Step 3.2
This simplifies to $\frac{4by^2 \cdot 9}{3y^3 \cdot 2b}$.
Step 4
Look for common factors in the numerator and denominator and cancel them out.
Step 4.1
Extract the common factor of $2$ from the numerator to get $\frac{2(2by^2 \cdot 9)}{3y^3 \cdot 2b}$.
Step 4.2
Proceed to cancel out the common factors.
Step 4.2.1
Extract the common factor of $2$ from the denominator to get $\frac{2(2by^2 \cdot 9)}{2(3y^3 \cdot b)}$.
Step 4.2.2
Cancel the common factor of $2$ to simplify to $\frac{2by^2 \cdot 9}{3y^3 \cdot b}$.
Step 5
Cancel out the common factor of $b$.
Step 5.1
Remove the common $b$ factor to get $\frac{2 \cancel{b} y^2 \cdot 9}{3y^3 \cancel{b}}$.
Step 5.2
This simplifies further to $\frac{2y^2 \cdot 9}{3y^3}$.
Step 6
Reduce the expression by canceling out common $y$ terms.
Step 6.1
Factor out $y^2$ from the numerator to get $\frac{y^2(2 \cdot 9)}{3y^3}$.
Step 6.2
Proceed to cancel out the common $y^2$ factors.
Step 6.2.1
Factor $y^2$ out of the denominator to get $\frac{y^2(2 \cdot 9)}{y^2(3y)}$.
Step 6.2.2
Cancel the common $y^2$ factor to simplify to $\frac{2 \cdot 9}{3y}$.
Step 7
Cancel out the common factor of $9$ and $3$.
Step 7.1
Factor out $3$ from the numerator to get $\frac{3(2 \cdot 3)}{3y}$.
Step 7.2
Proceed to cancel out the common $3$ factors.
Step 7.2.1
Factor $3$ out of the denominator to get $\frac{3(2 \cdot 3)}{3(y)}$.
Step 7.2.2
Cancel the common $3$ factor to simplify to $\frac{2 \cdot 3}{y}$.
Step 8
Finally, multiply $2$ by $3$ to get the simplified result $\frac{6}{y}$.
Knowledge Notes:
To simplify the division of fractions, the following knowledge points are relevant:
Reciprocal of a Fraction: The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$.
Multiplying Fractions: To multiply fractions, multiply the numerators together and the denominators together.
Simplifying Fractions: To simplify fractions, cancel out common factors from the numerator and denominator.
Exponents: When multiplying terms with the same base, add the exponents.
Canceling Common Factors: When a factor appears in both the numerator and the denominator, it can be canceled out.
Final Multiplication: After simplification, perform any remaining multiplications to get the final result.
These knowledge points are used in the steps outlined to simplify the given expression.
