Question

Explain how to get that answer!!

Answer 1

We need to simplify
`( √(14x^3) )/( √(18x) )`

First lets factor
`√(14x^3)`


`√(14x^3)` =
`√(14) √(x^3)`

`√(14) = √(2) √(7)` by applying the radical rule
`\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}`

`√(x^3) = x^(3/2)` By applying the radical rule
`\sqrt[n]{x^m} = x^(m/n)`

So

`√(14x^3)` =
`√(14) √(x^3)` =
`√(2) √(7)x^(3/2)`

Now let's factor
`√(18x)`
By applying the radical rule
`\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}`,

`√(18x) = √(18) √(x)`

`√(18) = √(2) * 3`

So
`√(18x)` =
`√(2)*3 √(x)`

So
`( √(14x^3) )/( √(18x) )` =
`( √(2) √(7) x^(3/2) )/( √(2)*3 √(x) )`

We know that
`\sqrt[n]{x} = x^(1/n)` so
`√(x) = x^(1/2)`

We now have
`( √(2) √(7) x^(3/2) )/( √(2)*3 √(x)) = ( √(2) √(7) x^(3/2) )/( √(2)*3x^(1/2))`

We know that
`(x^a)/(x^b) = x^(a-b)`
So
`(x^(3/2))/(x^(1/2)) = x^(3/2 - 1/2) = x`

We now got
`( √(2) √(7) x^(3/2) )/( √(2)*3x^(1/2)) = ( √(2) √(7) x )/( √(2)*3)`

We can notice that the numerator and the denominator both got √2 in a multiplication, so we can simplify them, and we get:

`( √(2) √(7) x )/( √(2)*3) = ( √(7)x )/(3)`


All in All, we get
`( √(14x^3) )/( √(18x) )` =
`( √(7)x )/(3)`

Hope this helps! :D