1. Rationalize the Denominator

can someone please show me howo to do this

^3 squarerootsign 5x / ^3 squarerootsign 4y

is the squreroot sign and radical? sign the same thing?

2. Originally Posted by rj2001
can someone please show me howo to do this

^3 squarerootsign 5x / ^3 squarerootsign 4y

is the squreroot sign and radical? sign the same thing?

The "squareroot sign" IS the radical (it's not actually called a "squareroot sign"). When the radical has no superscript written in front of it (such as the ^3 that you were writing), it's called a square root. With the ^3 in front, it is called a cubed root or the third root.

cubedroot(5x)/cubedroot(4y)

Now, to rationalize this, we need to multiply the numerator and denominator by something that will remove the cubed root on the denominator. Recall that to rationalize a square root we need to multiply the term by itself once; however, with cubed roots we need to multiply by the term twice or the term "squared"). To remove the cubed root on the denominator, we need to multiply the numerator and denominator by (cubedroot(4y))^2

[cubedroot(5x)*cubedroot(4y)^2]/[cubedroot(4y)*cubedroot(4y)^2]
cubedroot(5x*(4y)^2)/cubedroot(4y*(4y)^2)
cubedroot(80xy^2)/cubedroot(64y^3)

The cubed root of 80 = 2*cubedroot(10), and the cubed root of 64y^3 = 4y, so we get

2*cubedroot(10xy^2)/(4y) = cubedroot(10xy^2)/(2y)

I hope that makes sense

3. Hello, rj2001!

A slightly different approach . . .

. . . . . . . . . . . __ . . . __
Rationalize: .³√5x / ³√4y
We must make the denominator a cube.

. . . . . . . . . . . . . . . . . . . . . . ___
Multiply top and bottom by: .³√2y²

. . . . . . . . . . __ . . . .___ . . - . . ._____ . - . . . _____
. . . . . . . . ³√5x . . ³√2y² . . . . ³√10xy² . . . .³√10xy²
We have: . ------- · -------- . = . ---------- . = . ----------
. . . . . . . . ³√4y . . ³√2y² . . . . .³√8y³ . . . . . . . 2y