Question

which expression is equivalent to \\(\frac{2a + 1}{10a - 5} div \frac{10a}{4a^2 - 1}\\)?
\\(\bigcirc \frac{2a}{(2a - 1)^2}\\)
\\(\bigcirc \frac{50a}{(2a + 1)^2}\\)
\\(\bigcirc \frac{(2a - 1)^2}{2a}\\)
\\(\bigcirc \frac{(2a + 1)^2}{50a}\\)

Explanation:

Step1: Rewrite division as multiplication

To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. So, \(\frac{2a + 1}{10a - 5}\div\frac{10a}{4a^{2}-1}=\frac{2a + 1}{10a - 5}\times\frac{4a^{2}-1}{10a}\)

Step2: Factor the expressions

Factor \(10a - 5\): \(10a-5 = 5(2a - 1)\)
Factor \(4a^{2}-1\) (using the difference of squares formula \(x^{2}-y^{2}=(x + y)(x - y)\)): \(4a^{2}-1=(2a + 1)(2a - 1)\)
Substitute the factored forms into the expression: \(\frac{2a + 1}{5(2a - 1)}\times\frac{(2a + 1)(2a - 1)}{10a}\)

Step3: Cancel out common factors

Cancel out the common factors \((2a - 1)\) from the numerator and the denominator: \(\frac{2a + 1}{5}\times\frac{(2a + 1)}{10a}\)
Then multiply the numerators and the denominators: \(\frac{(2a + 1)(2a + 1)}{5\times10a}=\frac{(2a + 1)^{2}}{50a}\)

Answer:

\(\frac{(2a + 1)^{2}}{50a}\) (the fourth option: \(\boldsymbol{\frac{(2a + 1)^{2}}{50a}}\))