Simplify 12/((10x-40)/(4/(2x-8)))
The problem requires you to simplify the mathematical expression given, which involves fractions and variables. It is asking to resolve and reduce the complex fraction 12 divided by another fraction, which is (10x-40) divided by (4/(2x-8)). It is expected that you will carry out the necessary algebraic manipulations, which may include finding a common denominator, canceling out like terms, and performing operations to simplify the complex fraction to its simplest form.
$\frac{12}{\frac{10 x - 40}{\frac{4}{2 x - 8}}}$
Answer
Solution:
Step 1
Identify and eliminate common factors between $10x-40$ and $4/(2x-8)$.
Step 1.1
Extract the factor of $2$ from $4$.
$$\frac{12}{\frac{10x-40}{\frac{2 \cdot 2}{2x-8}}}$$
Step 1.2
Eliminate the common factors.
Step 1.2.1
Take $2$ out of $2x$.
$$\frac{12}{\frac{10x-40}{\frac{2 \cdot 2}{2(x)-8}}}$$
Step 1.2.2
Extract $2$ from $-8$.
$$\frac{12}{\frac{10x-40}{\frac{2 \cdot 2}{2(x)+2(-4)}}}$$
Step 1.2.3
Factor out $2$ from $2(x)+2(-4)$.
$$\frac{12}{\frac{10x-40}{\frac{2 \cdot 2}{2(x-4)}}}$$
Step 1.2.4
Remove the common factor.
$$\frac{12}{\frac{10x-40}{\frac{\cancel{2} \cdot 2}{\cancel{2}(x-4)}}}$$
Step 1.2.5
Reformulate the expression.
$$\frac{12}{\frac{10x-40}{2/(x-4)}}$$
Step 2
Invert the denominator and multiply it by the numerator.
$$12 \cdot \frac{2/(x-4)}{10x-40}$$
Step 3
Multiply the numerator by the inverted denominator.
$$12 \cdot \left( \frac{2}{x-4} \cdot \frac{1}{10x-40} \right)$$
Step 4
Factor out $10$ from $10x-40$.
Step 4.1
Extract $10$ from $10x$.
$$12 \cdot \left( \frac{2}{x-4} \cdot \frac{1}{10(x)-40} \right)$$
Step 4.2
Factor $10$ from $-40$.
$$12 \cdot \left( \frac{2}{x-4} \cdot \frac{1}{10x+10 \cdot -4} \right)$$
Step 4.3
Factor $10$ from $10x+10 \cdot -4$.
$$12 \cdot \left( \frac{2}{x-4} \cdot \frac{1}{10(x-4)} \right)$$
Step 5
Eliminate the common factor of $2$.
Step 5.1
Factor $2$ from $10(x-4)$.
$$12 \cdot \left( \frac{2}{x-4} \cdot \frac{1}{2 \cdot 5(x-4)} \right)$$
Step 5.2
Remove the common factor.
$$12 \cdot \left( \frac{\cancel{2}}{x-4} \cdot \frac{1}{\cancel{2} \cdot 5(x-4)} \right)$$
Step 5.3
Rephrase the expression.
$$12 \cdot \left( \frac{1}{x-4} \cdot \frac{1}{5(x-4)} \right)$$
Step 6
Combine $\frac{1}{x-4}$ with $\frac{1}{5(x-4)}$.
$$12 \cdot \frac{1}{(x-4) \cdot 5(x-4)}$$
Step 7
Raise $(x-4)$ to the power of $1$.
$$12 \cdot \frac{1}{5 \cdot ((x-4)^1 \cdot (x-4))}$$
Step 8
Raise $(x-4)$ to the power of $1$ again.
$$12 \cdot \frac{1}{5 \cdot ((x-4)^1 \cdot (x-4)^1)}$$
Step 9
Apply the exponent rule $a^m \cdot a^n = a^{m+n}$ to combine exponents.
$$12 \cdot \frac{1}{5 \cdot (x-4)^{1+1}}$$
Step 10
Add the exponents $1$ and $1$.
$$12 \cdot \frac{1}{5 \cdot (x-4)^2}$$
Step 11
Combine $12$ with $\frac{1}{5 \cdot (x-4)^2}$.
$$\frac{12}{5 \cdot (x-4)^2}$$
Knowledge Notes:
The problem involves simplifying a complex fraction by eliminating common factors and applying the properties of exponents. The steps include:
Factoring out common elements from the numerator and the denominator.
Multiplying by the reciprocal of the denominator.
Factoring out common numerical coefficients.
Applying exponent rules such as $a^m \cdot a^n = a^{m+n}$ to simplify expressions with the same base.
Simplifying the final expression by combining the constants and the simplified denominator.
This process requires knowledge of basic algebraic manipulation, including factoring, working with fractions, and exponent rules.
