Simplify ((4by)/(3y^3))÷((2b)/(9y))

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 $\frac{4b{y}^2\cdot 9}{3y^3\cdot 2b}$.

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 $\frac{2(2b{y}^2\cdot 9)}{3y^3\cdot 2b}$.

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 $\frac{2(2b{y}^2\cdot 9)}{2(3y^3\cdot b)}$.

Step 4.2.2

Cancel the common factor of $2$ to simplify to $\frac{2b{y}^2\cdot 9}{3y^3\cdot b}$.

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:

1. Reciprocal of a Fraction: The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$.

2. Multiplying Fractions: To multiply fractions, multiply the numerators together and the denominators together.

3. Simplifying Fractions: To simplify fractions, cancel out common factors from the numerator and denominator.

4. Exponents: When multiplying terms with the same base, add the exponents.

5. Canceling Common Factors: When a factor appears in both the numerator and the denominator, it can be canceled out.

6. 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.