Simplify (((y-7)^2)/12)/((12y-84)/144)
The given problem is a mathematical question involving the simplification of a complex rational expression. The expression consists of a numerator, which is a binomial (y-7) squared and then divided by 12, and a denominator, which is a linear expression (12y-84) divided by 144. The task is to simplify this complex fraction by performing the necessary arithmetic operations, including factoring, canceling common factors, and simplifying the resulting expression to its simplest form.
$\frac{\frac{\left(\left(\right. y - 7 \left.\right)\right)^{2}}{12}}{\frac{12 y - 84}{144}}$
Answer
Solution:
Step 1
Identify and remove the common factors between $12y - 84$ and $144$.
Step 1.1
Extract the factor of $12$ from $12y$ in the denominator.
$$\frac{\frac{(y - 7)^2}{12}}{\frac{12(y) - 84}{144}}$$
Step 1.2
Extract the factor of $12$ from $-84$ in the denominator.
$$\frac{\frac{(y - 7)^2}{12}}{\frac{12y - 12 \cdot 7}{144}}$$
Step 1.3
Combine the factored terms in the denominator.
$$\frac{\frac{(y - 7)^2}{12}}{\frac{12(y - 7)}{144}}$$
Step 1.4
Eliminate the common factors.
Step 1.4.1
Factor out $12$ from $144$ in the denominator.
$$\frac{\frac{(y - 7)^2}{12}}{\frac{12(y - 7)}{12 \cdot 12}}$$
Step 1.4.2
Cancel out the common factor of $12$.
$$\frac{\frac{(y - 7)^2}{12}}{\frac{\cancel{12}(y - 7)}{\cancel{12} \cdot 12}}$$
Step 1.4.3
Simplify the expression.
$$\frac{\frac{(y - 7)^2}{12}}{\frac{y - 7}{12}}$$
Step 2
Multiply the numerator by the reciprocal of the denominator.
$$\frac{(y - 7)^2}{12} \cdot \frac{12}{y - 7}$$
Step 3
Remove the common factor of $y - 7$.
Step 3.1
Factor out $y - 7$ from $(y - 7)^2$ in the numerator.
$$\frac{(y - 7)(y - 7)}{12} \cdot \frac{12}{y - 7}$$
Step 3.2
Cancel out the common factor of $y - 7$.
$$\frac{\cancel{(y - 7)}(y - 7)}{12} \cdot \frac{12}{\cancel{y - 7}}$$
Step 3.3
Simplify the expression.
$$\frac{y - 7}{12} \cdot 12$$
Step 4
Eliminate the common factor of $12$.
Step 4.1
Cancel out the common factor of $12$.
$$\frac{y - 7}{\cancel{12}} \cdot \cancel{12}$$
Step 4.2
Simplify the expression to get the final result.
$$y - 7$$
Knowledge Notes:
The problem involves simplifying a complex fraction by identifying and canceling common factors. The steps taken include:
Factoring out common terms in the numerator and denominator.
Recognizing that division by a fraction is equivalent to multiplication by its reciprocal.
Canceling out common factors in both the numerator and the denominator to simplify the expression.
Understanding that when a term is squared, it can be factored into the term multiplied by itself.
Key concepts used in this problem include:
Factoring expressions to reveal common factors.
The property of reciprocals, which states that the reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$.
The cancellation property, which allows us to simplify expressions by removing common factors from the numerator and denominator.
In this problem, we applied these concepts to simplify the given complex fraction to its simplest form.
