Simplify the following.  √(128x^6)+√(98x^6)

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embizze | High School Teacher | (Level 1) Educator Emeritus

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A radical expression is simplified if there are no perfect nth powers under the radical, where n is the index. (Also no radicals in the denominator or fractions under the radical.)

Simplify `sqrt(128x^6)+sqrt(98x^6)`

First look for perfect square factors of the radicand:

`sqrt(128x^6)+sqrt(98x^6)`

`=sqrt(64x^6*2)+sqrt(49x^6*2)` Then use `sqrt(ab)=sqrt(a)sqrt(b)` :

`=sqrt(64x^6)sqrt(2)+sqrt(49x^6)sqrt(2)`

`=8x^3sqrt(2)+7x^3sqrt(2)` Add like terms:

`=15x^3sqrt(2)` which is simplified.

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I like to check the answer for some x: let x=3;

`sqrt(128*3^6)+sqrt(98*3^6)=216sqrt(2)+189sqrt(2)=405sqrt(2)=15*3^3sqrt(2)`

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

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When you see problems like this, go to the instructions to see if there are restrictions on what `x` can be. Most books I've seen specify that all variables are non-negative, for the following reason.

Let `x=-3.` You'll find that `sqrt(128x^6)+sqrt(98x^6)=405sqrt(2),`

while `15x^3sqrt(2)=-405sqrt(2),`

so the expressions are not equal. The reason is that `sqrt(x^6)=x^3` is not true if `x` is negative. What is true for all values of `x` is that `sqrt(x^6)=|x|^3,` but I guess constantly having to write things like `15|x|^3sqrt(2)` (or simply leaving `sqrt(x^6)` unchanged) is deemed awkward, so many books make the assumption of non-negative variables.

Also, at the risk of overloading this post with details, some posters here use the convention that `sqrt(x^6)=+-x^3,` so their square root function is multivalued. I've never seen that in any book (and I doubt yours uses that convention), but I guess they're out there, and they add another layer to the problem.

m-i-s-h-a-l's profile pic

m-i-s-h-a-l | eNotes Newbie

Posted on

x=3

schoolkid's profile pic

schoolkid | eNotes Newbie

Posted on

A radical expression is simplified if there are no perfect nth powers under the radical, where n is the index. (Also no radicals in the denominator or fractions under the radical.)

Simplify 

First look for perfect square factors of the radicand:

 Then use  :

 Add like terms:

 which is simplified.

-----------------------------------------------------

I like to check the answer for some x: let x=3;

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