Problem

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.

x2956x3x7xy

Answer

Expert–verified

Solution:

Step:1

Multiply the numerator by the reciprocal of the denominator.

x2956x7xy3x

Step:2

Simplify terms.

Step:2.1

Eliminate the common factor of 7x.

Step:2.1.1

Extract 7x from 56x.

x297x87xy3x

Step:2.1.2

Extract 7x from 7xy.

x297x87xy3x

Step:2.1.3

Remove the common factor.

x297x87xy3x

Step:2.1.4

Reformulate the expression.

x298y3x

Step:2.2

Combine x298 with y3x.

(x29)y8(3x)

Step:3

Refactor the numerator.

Step:3.1

Represent 9 as 32.

(x232)y8(3x)

Step:3.2

Apply the difference of squares formula, a2b2=(a+b)(ab), with a=x and b=3.

(x+3)(x3)y8(3x)

Step:4

Cancel out the common factors of x3 and 3x.

Step:4.1

Factor out 1 from x.

(x+3)(1(x3))y8(3x)

Step:4.2

Express 3 as 1(3).

(x+3)(1(x1(3)))y8(3x)

Step:4.3

Factor 1 from 1(x1(3)).

(x+3)(1(x+3))y8(3x)

Step:4.4

Rearrange the terms.

(x+3)(1(x+3))y8(x+3)

Step:4.5

Eliminate the common factor.

(x+3)(1(x+3))y8(x+3)

Step:4.6

Rephrase the expression.

((x+3)(1))y8

Step:5

Extract the negative sign.

(x+3)y8

Step:6

Place the negative sign in front of the fraction.

(x+3)y8

Step:7

Rearrange the factors in (x+3)y8.

y(x+3)8

Knowledge Notes:

To simplify the given complex fraction, we follow these steps:

  1. Multiplication by Reciprocal: To divide by a fraction, we multiply by its reciprocal.

  2. Simplification: We look for common factors in the numerator and denominator to cancel out.

  3. Difference of Squares: This is a technique used to factor expressions of the form a2b2 into (a+b)(ab).

  4. Negative Signs: Factoring out negative signs can help to simplify expressions and reveal common factors.

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

  6. Rearrangement: Sometimes, rearranging terms can make it easier to identify and cancel out common factors.

  7. 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.

link_gpt