Given the expression,
[tex] (8x-56x^2)/(\sqrt{14x-2}) [/tex]
We will have to rationalize the denominator first. To rationalize the denominator we have to multiply the numerator and denominator both by the square root part of the denominator.
[tex] [(8x-56x^2)(\sqrt{14x-2})]/[(\sqrt{14x-2})(\sqrt{14x-2})] [/tex]
If we have [tex] (\sqrt{a})(\sqrt{a}) [/tex], we will get [tex] \sqrt{a^2} [/tex] by multiplying them. And [tex] \sqrt{a^2} = a [/tex].
So here in the problem, we will get,
[tex] [(8x-56x^2)(\sqrt{14x-2})]/(14x-2) [/tex]
Now in the numerator we have [tex] (8x-56x^2) [/tex]. We can check 8x is common there. we will take out -8x from it, we will get,
[tex] -8x(-1+7x) [/tex]
[tex] -8x(7x-1) [/tex]
And in the denominator we have [tex] 14x-2 [/tex]. We can check 2 is common there. If we take out 2 from it we will get,
[tex] 2(7x-1) [/tex]
So we can write the expression as
[tex] [(-8x)(7x-1)(\sqrt{14x-2})]/[2(7x-1)] [/tex]
[tex] (7x-1) [/tex] is common to the numerator and denominator both, if we cancel it we will get,
[tex] (-8x)(\sqrt{14x-2})/2 [/tex]
We can divide -8 by the denominator, as -8 os divisible by 2. By dividing them we will get,
[tex] (-4x)(\sqrt{14x-2}) [/tex]
[tex] (-4x)(14x-2)^(1/2) [/tex]
So we have got the required answer here.
The correct option is the last one.
