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((x^2-14x+49)/(x^2-49))/((3x-21)/(x+7))

3 min read Jun 16, 2024
((x^2-14x+49)/(x^2-49))/((3x-21)/(x+7))

Simplifying the Expression: ((x^2-14x+49)/(x^2-49))/((3x-21)/(x+7))

This article will guide you through the process of simplifying the given rational expression.

Understanding the Expression

The expression is a complex fraction, meaning it has a fraction in the numerator and another in the denominator. To simplify this, we can follow these steps:

  1. Factorize the Expressions:

    • The numerator of the main fraction: (x^2 - 14x + 49) is a perfect square trinomial, which factors as (x - 7)^2.
    • The denominator of the main fraction: (x^2 - 49) is a difference of squares, which factors as (x + 7)(x - 7).
    • The numerator of the denominator fraction: (3x - 21) can be factored by taking out the common factor 3, resulting in 3(x - 7).

  2. Rewrite the Expression:

    Now our expression looks like this: [(x - 7)^2 / ((x + 7)(x - 7))] / [3(x - 7) / (x + 7)]

  3. Divide Fractions:

    Dividing fractions is the same as multiplying by the reciprocal of the second fraction. This means we can rewrite the expression as: [(x - 7)^2 / ((x + 7)(x - 7))] * [(x + 7) / 3(x - 7)]

  4. Cancel Common Factors:

    Notice that (x - 7) appears in the numerator and denominator, and (x + 7) also appears in the numerator and denominator. Canceling these common factors simplifies the expression: (x - 7) / 3

Final Simplified Expression

Therefore, the simplified form of the expression ((x^2-14x+49)/(x^2-49))/((3x-21)/(x+7)) is (x - 7) / 3.

Important Note: The expression is undefined when x = 7 or x = -7, as these values would make the denominator of the original fractions equal to zero.

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