Simplify (a/b-c/d)/(b/a-d/c)
The question is asking you to perform algebraic simplification on a complex fraction. The fraction consists of two parts: the numerator, which is (a/b - c/d), and the denominator, which is (b/a - d/c). The task is to simplify this expression into its simplest form by performing operations such as finding common denominators, reducing fractions, and simplifying the overall expression possibly by canceling out terms if applicable.
$\frac{\frac{a}{b} - \frac{c}{d}}{\frac{b}{a} - \frac{d}{c}}$
Answer
Solution:
Step:1
Multiply both the numerator and the denominator by $acbd$.
Step:1.1
Multiply $\frac{a/b - c/d}{b/a - d/c}$ by $\frac{acbd}{acbd}$ to get $\frac{acbd}{acbd} \cdot \frac{a/b - c/d}{b/a - d/c}$.
Step:1.2
Combine the terms to get $\frac{acbd(a/b - c/d)}{acbd(b/a - d/c)}$.
Step:2
Distribute $acbd$ across both the numerator and the denominator.
Step:3
Cancel out common factors.
Step:3.1
Eliminate the common factor $b$.
Step:3.1.1
Extract $b$ from $acbd$ to get $\frac{b(dac)(a/b) - b(dac)(c/d)}{b(dac)(b/a) - b(dac)(d/c)}$.
Step:3.1.2
Cancel out $b$ to simplify to $\frac{dac(a) - dac(c)}{dac(b) - dac(d)}$.
Step:3.2 to Step:3.15
Repeat the process of raising to the power of 1, combining exponents, and canceling common factors for $a$, $c$, and $d$.
Step:4
Simplify the numerator by factoring out $ac$.
Step:4.1
Factor $ac$ from $d(a^2)c + b(-c^2)a$ to get $\frac{ac(da - bc)}{remainder}$.
Step:5
Simplify the denominator by factoring out $bd$.
Step:5.1
Factor $bd$ from $b^2(dc) + b(-d^2)a$ to get $\frac{remainder}{bd(bc - ad)}$.
Step:6
Reduce the fraction by canceling out the common factor $ad - bc$.
Step:6.1 to Step:6.3
Reorder terms, factor out $-1$, and cancel the common factor to simplify to $-\frac{ac}{bd}$.
Knowledge Notes:
Multiplying by a form of 1: Multiplying a fraction by a fraction that is equivalent to 1 (like $\frac{acbd}{acbd}$) does not change its value but can simplify the expression.
Distributive Property: This property allows us to multiply a single term by each term inside a set of parentheses.
Canceling Common Factors: If a factor appears in both the numerator and the denominator of a fraction, it can be canceled out.
Raising to the Power of 1: Any number raised to the power of 1 is itself. This is sometimes used to make the format of terms consistent for simplification.
Combining Exponents (Power Rule): When multiplying like bases, add the exponents ($a^m \cdot a^n = a^{m+n}$).
Factoring: This involves taking common factors out of terms to simplify expressions.
Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not change the product ($ab = ba$).
Negative Factors: A negative factor can be moved across terms in a numerator or between the numerator and denominator.
