Multiply (6a^2)/(5b^2)*(45b^3)/(18a^3)

Solution:

Step 1:

Combine the fractions to form a single expression.

$$\frac{6a^2 \cdot 45b^3}{5b^2 \cdot 18a^3}$$

Step 2:

Reduce the fraction by eliminating common factors between numerator and denominator.

Step 2.1:

Identify and factor out the common factor between 6 and 18 in the numerator and denominator.

$$\frac{6(a^2 \cdot 45b^3)}{5b^2 \cdot 18a^3}$$

Step 2.2:

Proceed to simplify by canceling out the common factor.

Step 2.2.1:

Factor out the common factor of 6 in the denominator.

$$\frac{6(a^2 \cdot 45b^3)}{6(5b^2 \cdot 3a^3)}$$

Step 2.2.2:

Eliminate the common factor of 6.

$$\frac{\cancel{6}(a^2 \cdot 45b^3)}{\cancel{6}(5b^2 \cdot 3a^3)}$$

Step 2.2.3:

Simplify the expression after cancelation.

$$\frac{a^2 \cdot 45b^3}{5b^2 \cdot 3a^3}$$

Step 3:

Further reduce the expression by canceling out any additional common factors.

Step 3.1:

Factor out the common factor of $a^2$ in the denominator.

$$\frac{a^2 \cdot 45b^3}{a^2(5b^2 \cdot 3a)}$$

Step 3.2:

Cancel the common factor of $a^2$.

$$\frac{\cancel{a^2} \cdot 45b^3}{\cancel{a^2}(5b^2 \cdot 3a)}$$

Step 3.3:

Simplify the expression after cancelation.

$$\frac{45b^3}{5b^2 \cdot 3a}$$

Step 4:

Continue simplifying by canceling out common numerical factors.

Step 4.1:

Factor out the common factor of 5 in the numerator.

$$\frac{5 \cdot 9b^3}{5b^2 \cdot 3a}$$

Step 4.2:

Proceed to simplify by canceling out the common factor.

Step 4.2.1:

Factor out the common factor of 5 in the denominator.

$$\frac{5 \cdot 9b^3}{5(b^2 \cdot 3a)}$$

Step 4.2.2:

Eliminate the common factor of 5.

$$\frac{\cancel{5} \cdot 9b^3}{\cancel{5}(b^2 \cdot 3a)}$$

Step 4.2.3:

Simplify the expression after cancelation.

$$\frac{9b^3}{b^2 \cdot 3a}$$

Step 5:

Reduce the expression by canceling out common numerical factors.

Step 5.1:

Factor out the common factor of 3 in the numerator.

$$\frac{3 \cdot 3b^3}{b^2 \cdot 3a}$$

Step 5.2:

Proceed to simplify by canceling out the common factor.

Step 5.2.1:

Factor out the common factor of 3 in the denominator.

$$\frac{3 \cdot 3b^3}{3(b^2 \cdot a)}$$

Step 5.2.2:

Eliminate the common factor of 3.

$$\frac{\cancel{3} \cdot 3b^3}{\cancel{3}(b^2 \cdot a)}$$

Step 5.2.3:

Simplify the expression after cancelation.

$$\frac{3b^3}{b^2 \cdot a}$$

Step 6:

Finalize the simplification by canceling out any remaining common factors.

Step 6.1:

Factor out the common factor of $b^2$ in the numerator.

$$\frac{b^2 \cdot 3b}{b^2 \cdot a}$$

Step 6.2:

Cancel the common factors.

Step 6.2.1:

Eliminate the common factor of $b^2$.

$$\frac{\cancel{b^2} \cdot 3b}{\cancel{b^2} \cdot a}$$

Step 6.2.2:

Simplify the expression after cancelation.

$$\frac{3b}{a}$$

Knowledge Notes:

To solve the given problem, we apply the following knowledge points:

1. Multiplication of Fractions: To multiply fractions, we multiply the numerators together and the denominators together.

2. Simplifying Fractions: This involves canceling out common factors between the numerator and the denominator.

3. Factoring: This is the process of breaking down numbers into their constituent prime factors or common factors.

4. Cancelation: When a factor appears in both the numerator and the denominator, it can be canceled out from both, simplifying the fraction.

5. Exponent Rules: When we have the same base with exponents in both the numerator and the denominator, we can subtract the exponents if we are dividing.

6. LaTeX Formatting: Mathematical expressions are formatted using LaTeX syntax to clearly present the problem-solving process.

By applying these principles, we can simplify the given algebraic fraction to its simplest form.