Simplify ((x^2-4)/(x-2))÷((x+2)/(4x-8))
The given problem is a mathematical expression that involves simplification. The task is to simplify the complex fraction or expression where the numerator is a binomial difference (x^2-4) over (x-2), and the denominator of the division is another binomial (x+2) over (4x-8). The expression takes the form of a division of two fractions, and it requires applying the rules of algebraic manipulation, such as factorization, simplifying binomials, recognizing difference of squares, and division of fractions by multiplying by the reciprocal, to obtain a simpler form or possibly a single rational expression.
$\frac{x^{2} - 4}{x - 2} \div \frac{x + 2}{4 x - 8}$
Answer
Solution:
Step:1
To perform division with fractions, switch to multiplication by the inverse of the divisor. $\frac{x^{2} - 4}{x - 2} \cdot \frac{4x - 8}{x + 2}$
Step:2
Begin by simplifying the numerator.
Step:2.1
Express $4$ as $2^{2}$. $\frac{x^{2} - 2^{2}}{x - 2} \cdot \frac{4x - 8}{x + 2}$
Step:2.2
Utilize the difference of squares identity, $a^{2} - b^{2} = (a + b)(a - b)$, with $a = x$ and $b = 2$. $\frac{(x + 2)(x - 2)}{x - 2} \cdot \frac{4x - 8}{x + 2}$
Step:3
Simplify the expression by eliminating common factors.
Step:3.1
Remove the shared factor of $x + 2$.
Step:3.1.1
Eliminate the shared factor. $\frac{(\cancel{x + 2})(x - 2)}{x - 2} \cdot \frac{4x - 8}{\cancel{x + 2}}$
Step:3.1.2
Reformulate the expression. $\frac{x - 2}{x - 2} \cdot (4x - 8)$
Step:3.2
Eliminate the shared factor of $x - 2$.
Step:3.2.1
Remove the shared factor. $\frac{\cancel{x - 2}}{\cancel{x - 2}} \cdot (4x - 8)$
Step:3.2.2
Rephrase the expression. $1 \cdot (4x - 8)$
Step:3.3
Multiply $1$ by $(4x - 8)$. $4x - 8$
Knowledge Notes:
To simplify the given expression, we use the following knowledge points:
Division of Fractions: To divide one fraction by another, multiply the first fraction by the reciprocal of the second. This means that $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$.
Difference of Squares: This is a pattern for factoring expressions of the form $a^2 - b^2$, which can be rewritten as $(a + b)(a - b)$.
Factoring: This involves rewriting an expression as a product of its factors. It is often used to simplify expressions by canceling out common factors in the numerator and denominator.
Cancellation: If a factor appears in both the numerator and denominator of a fraction, it can be canceled out, since dividing by a number is the same as multiplying by its reciprocal.
Simplification: After canceling common factors, the expression may often be simplified further to a more basic form, which is the goal of the problem-solving process.
By applying these concepts, we can simplify the given complex fraction to its simplest form.
