Rationalize the following expression 1 / (2³√a + 3³√b)

Please show solution with an explanation.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

You need to remove the radicals from denominator, hence you need to remember the following formula such that:

`x^3 + y^3 = (x + y)(x^2 - xy + y^2)`

Notice that denominator only contains the `(x+y)`  part and you need to multiply by `(x^2 - xy + y^2)`  part to remove the cube root.

You need to substitute `2root(3)a`  for x and `3root(3)b`  for y and you need to multiply both numerator and denominator by `(4root(3)(a^2) - 6root(3)(ab) + 9root(3)(b^2))`  such that:

`(4root(3)(a^2) - 6root(3)(ab) + 9root(3)(b^2))/((2root(3)a + 3root(3)b)(4root(3)(a^2)- 6root(3)(ab) + 9root(3)(b^2))) = (4root(3)(a^2) - 6root(3)(ab) + 9root(3)(b^2))/(8a+ 27b)`

Hence, rationalizing the fraction yields `1/(2root(3)a + 3root(3)b) = (4root(3)(a^2) - 6root(3)(ab) + 9root(3)(b^2))/(8a + 27b).`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial