Simplify ((u^2-9)/(25u+75))/((5u^2-30u+45)/(125u+375))
The problem you are presented with is a complex rational expression that requires simplification. A rational expression is a fraction in which the numerator and the denominator are polynomials.
This question asks you to simplify the complex fraction by performing appropriate mathematical operations to reduce the expression to its simplest form. Simplification typically involves factoring the polynomials where possible, canceling common factors in the numerator and denominator, and performing division of the polynomials by rewriting the complex fraction as a division problem and then multiplying by the reciprocal. The aim is to express the result in the simplest form with the polynomials in their most reduced state.
$\frac{\frac{u^{2} - 9}{25 u + 75}}{\frac{5 u^{2} - 30 u + 45}{125 u + 375}}$
Answer
Solution:
Step 1: Simplify the common factors in the numerator and denominator.
Step 1.1: Extract the factor of 5 from the terms in the denominator.
$$\frac{\frac{u^2 - 9}{25u + 75}}{\frac{5(u^2) - 5(6u) + 5(9)}{5(25u) + 5(75)}}$$
Step 1.2: Simplify the extracted factor of 5.
$$\frac{\frac{u^2 - 9}{25u + 75}}{\frac{5(u^2 - 6u + 9)}{5(25u + 75)}}$$
Step 1.3: Cancel out the factor of 5 in the denominator.
$$\frac{\frac{u^2 - 9}{25u + 75}}{\frac{u^2 - 6u + 9}{25u + 75}}$$
Step 2: Multiply the numerator by the reciprocal of the denominator.
$$\frac{u^2 - 9}{25u + 75} \cdot \frac{25u + 75}{u^2 - 6u + 9}$$
Step 3: Eliminate the common factor in the numerator and denominator.
Step 3.1: Cancel the common term $25u + 75$.
$$\frac{u^2 - 9}{u^2 - 6u + 9}$$
Step 4: Factor both the numerator and the denominator.
Step 4.1: Recognize the difference of squares in the numerator.
$$\frac{(u + 3)(u - 3)}{u^2 - 6u + 9}$$
Step 4.2: Factor the denominator as a perfect square.
$$\frac{(u + 3)(u - 3)}{(u - 3)^2}$$
Step 5: Simplify the fraction by canceling out common terms.
Step 5.1: Cancel the common term $(u - 3)$.
$$\frac{u + 3}{u - 3}$$
Knowledge Notes:
Factorization: The process of breaking down expressions into products of simpler expressions. Common factorization techniques include taking out common factors and using formulas such as the difference of squares $a^2 - b^2 = (a + b)(a - b)$ and perfect square trinomials $a^2 \pm 2ab + b^2 = (a \pm b)^2$.
Simplifying Complex Fractions: Involves finding a common denominator, if necessary, and then multiplying the numerator by the reciprocal of the denominator.
Canceling Common Factors: When a factor appears in both the numerator and the denominator of a fraction, it can be canceled out, simplifying the expression.
Difference of Squares: A special factoring technique used when an expression is in the form of $a^2 - b^2$. It can be factored into $(a + b)(a - b)$.
Perfect Square Trinomial: A trinomial of the form $a^2 \pm 2ab + b^2$ which factors into $(a \pm b)^2$.
Reciprocal: The reciprocal of a number (or expression) is 1 divided by that number (or expression). For example, the reciprocal of $x$ is $\frac{1}{x}$.
LaTeX Formatting: A typesetting system that is widely used for mathematical and scientific documents due to its ability to render complex formulas and expressions clearly.
