Simplify ((x^2-9)/(56x))/((3-x)/(7xy))
The question is asking you to simplify a complex rational expression, which is a fraction divided by another fraction. Both the numerator and the denominator of the overall expression contain algebraic expressions involving the variable x. The task is to perform the appropriate mathematical operations, such as factoring, multiplying, and dividing, to simplify the overall expression to its simplest form which may involve combining like terms or reducing any common factors between the numerator and denominator.
$\frac{\frac{x^{2} - 9}{56 x}}{\frac{3 - x}{7 x y}}$
Answer
Solution:
Step:1
Multiply the numerator by the reciprocal of the denominator.
$$\frac{x^2 - 9}{56x} \cdot \frac{7xy}{3 - x}$$
Step:2
Simplify terms.
Step:2.1
Eliminate the common factor of $7x$.
Step:2.1.1
Extract $7x$ from $56x$.
$$\frac{x^2 - 9}{7x \cdot 8} \cdot \frac{7xy}{3 - x}$$
Step:2.1.2
Extract $7x$ from $7xy$.
$$\frac{x^2 - 9}{7x \cdot 8} \cdot \frac{7x \cdot y}{3 - x}$$
Step:2.1.3
Remove the common factor.
$$\frac{x^2 - 9}{\cancel{7x} \cdot 8} \cdot \frac{\cancel{7x} y}{3 - x}$$
Step:2.1.4
Reformulate the expression.
$$\frac{x^2 - 9}{8} \cdot \frac{y}{3 - x}$$
Step:2.2
Combine $\frac{x^2 - 9}{8}$ with $\frac{y}{3 - x}$.
$$\frac{(x^2 - 9) y}{8(3 - x)}$$
Step:3
Refactor the numerator.
Step:3.1
Represent $9$ as $3^2$.
$$\frac{(x^2 - 3^2) y}{8(3 - x)}$$
Step:3.2
Apply the difference of squares formula, $a^2 - b^2 = (a + b)(a - b)$, with $a = x$ and $b = 3$.
$$\frac{(x + 3)(x - 3) y}{8(3 - x)}$$
Step:4
Cancel out the common factors of $x - 3$ and $3 - x$.
Step:4.1
Factor out $-1$ from $x$.
$$\frac{(x + 3)(-1(-x - 3)) y}{8(3 - x)}$$
Step:4.2
Express $-3$ as $-1(3)$.
$$\frac{(x + 3)(-1(-x - 1(3))) y}{8(3 - x)}$$
Step:4.3
Factor $-1$ from $-1(-x - 1(3))$.
$$\frac{(x + 3)(-1(-x + 3)) y}{8(3 - x)}$$
Step:4.4
Rearrange the terms.
$$\frac{(x + 3)(-1(-x + 3)) y}{8(-x + 3)}$$
Step:4.5
Eliminate the common factor.
$$\frac{(x + 3)(-1(\cancel{-x + 3})) y}{8(\cancel{-x + 3})}$$
Step:4.6
Rephrase the expression.
$$\frac{((x + 3) \cdot (-1)) y}{8}$$
Step:5
Extract the negative sign.
$$\frac{- (x + 3) y}{8}$$
Step:6
Place the negative sign in front of the fraction.
$$- \frac{(x + 3) y}{8}$$
Step:7
Rearrange the factors in $- \frac{(x + 3) y}{8}$.
$$- \frac{y(x + 3)}{8}$$
Knowledge Notes:
To simplify the given complex fraction, we follow these steps:
Multiplication by Reciprocal: To divide by a fraction, we multiply by its reciprocal.
Simplification: We look for common factors in the numerator and denominator to cancel out.
Difference of Squares: This is a technique used to factor expressions of the form $a^2 - b^2$ into $(a + b)(a - b)$.
Negative Signs: Factoring out negative signs can help to simplify expressions and reveal common factors.
Cancellation: When a factor appears in both the numerator and the denominator, it can be canceled out.
Rearrangement: Sometimes, rearranging terms can make it easier to identify and cancel out common factors.
Final Expression: The goal is to write the expression in its simplest form, which may involve moving negative signs or rearranging factors for clarity.
In this problem, we applied these principles to simplify the complex fraction step by step.
