Multiply (4x)/(ab)*(a^2b)/(2x^2)
The question is asking you to perform a multiplication operation between two rational expressions. Specifically, it requires you to multiply a fraction that has '4x' in the numerator and 'ab' in the denominator by another fraction with 'a^2b' in the numerator and '2x^2' in the denominator. You are expected to simplify the resulting expression by cancelling out any common factors in the numerator and denominator.
$\frac{4 x}{a b} \cdot \frac{a^{2} b}{2 x^{2}}$
Answer
Solution:
Step 1
Combine the fractions: $\frac{4x}{ab} \cdot \frac{a^2b}{2x^2}$.
Step 2
Simplify by reducing the coefficients 4 and 2.
Step 2.1
Extract the factor of 2 from $4x \cdot a^2b$: $\frac{2(2xa^2b)}{ab(2x^2)}$.
Step 2.2
Eliminate the common factors.
Step 2.2.1
Extract the factor of 2 from $ab \cdot 2x^2$: $\frac{2(2xa^2b)}{2(abx^2)}$.
Step 2.2.2
Remove the common factor of 2: $\frac{\cancel{2}(2xa^2b)}{\cancel{2}(abx^2)}$.
Step 2.2.3
Rewrite the simplified expression: $\frac{2xa^2b}{abx^2}$.
Step 3
Reduce the common variables $x$ and $x^2$.
Step 3.1
Extract the factor of x from $2xa^2b$: $\frac{x(2a^2b)}{abx^2}$.
Step 3.2
Eliminate the common factors.
Step 3.2.1
Extract the factor of x from $abx^2$: $\frac{x(2a^2b)}{xabx}$.
Step 3.2.2
Remove the common factor of x: $\frac{\cancel{x}(2a^2b)}{\cancel{x}(abx)}$.
Step 3.2.3
Rewrite the simplified expression: $\frac{2a^2b}{abx}$.
Step 4
Reduce the common factors $a^2$ and $a$.
Step 4.1
Extract the factor of a from $2a^2b$: $\frac{a(2ab)}{abx}$.
Step 4.2
Eliminate the common factors.
Step 4.2.1
Extract the factor of a from $abx$: $\frac{a(2ab)}{abx}$.
Step 4.2.2
Remove the common factor of a: $\frac{\cancel{a}(2ab)}{\cancel{a}(bx)}$.
Step 4.2.3
Rewrite the simplified expression: $\frac{2ab}{bx}$.
Step 5
Cancel out the common factor $b$.
Step 5.1
Remove the common factor of b: $\frac{2a\cancel{b}}{\cancel{b}x}$.
Step 5.2
Rewrite the final simplified expression: $\frac{2a}{x}$.
Knowledge Notes:
The problem involves simplifying a complex fraction by multiplying two fractions together. The process of simplification includes:
Combining Fractions: When multiplying fractions, we combine the numerators and the denominators respectively.
Reducing Coefficients: Coefficients are the numerical parts of the terms. If there are common factors in the coefficients of the numerator and denominator, we can simplify by dividing them by their greatest common divisor.
Cancelling Common Factors: If the same factor appears in both the numerator and the denominator, it can be cancelled out. This applies to both numerical coefficients and variables.
Rewriting Expressions: After simplification, it's important to rewrite the expression to reflect the reduced form.
Algebraic Manipulation: This includes factoring out common variables or coefficients and cancelling them out to simplify the expression.
LaTeX Formatting: Mathematical expressions are formatted using LaTeX to clearly display the equations and the simplification process.
In this problem, the process involves reducing the coefficients (4 and 2), cancelling common variables ($x$ and $x^2$), and then cancelling common algebraic terms ($a^2$ and $a$, $b$). The final result is a simplified fraction.
