Question

2x^(2/3)-x^(1/3)-1=0

Answer 1

\(\normalsize\color{black}{ 2x^{\LARGE\bf ⅔}-x^{\LARGE\bf ⅓}-1=0\LARGE\color{white}{ \rm │ }}\)

\(\normalsize\color{black}{ 2\sqrt[3]{x^2}-\sqrt[3]{x}-1=0\LARGE\color{white}{ \rm │ }}\)

\(\normalsize\color{black}{ 2\sqrt[3]{x^2}-\sqrt[3]{x}=1\LARGE\color{white}{ \rm │ }}\)

I think then you would cbe troot both sides.

Answer 2

*cube root both sides.

Answer 3

would you cube root or cube to remove the roots?

Answer 4

yeah, I mean to raise both sides to the 3rd power..

Answer 5

can you tell me what x would be equal to?

Answer 6

yes, it is 1

Answer 7

if you plug 1 into the equation as x, it's not equal to 0

Answer 8

yes it is

Answer 9

2x^(2/3)-x^(1/3)-1=0
2(1)^(2/3)-(1)^(1/3)-1=0
2(1)-(1)-1=0
2 - 1 - 1 =0
0=0

Answer 10

ah okay can you explain how i get x equals 1 after i raise each side to the third power ?

Answer 11

whatever you do, you don't change the value of the sides... right ?

Answer 12

notice if you let
y= x^(⅓)
then
\[ 2x^{⅔}-x^{⅓}-1=0 \\ 2y^2 -y -1 = 0 \]
solve for y
you can use the quadratic formula, or you can factor to find
(2y +1)(y -1) =0

you get two answers for y. you can use them in
y= x^(⅓) so x=y^3
to get the values for x

Answer 13

Thanks Phi. x is equal to -1 and +1

Answer 14

no, rethink that

Answer 15

(2y +1)(y -1) =0
2y+1=0 y-1=0
y+1=0 y-1=0
y=-1 y=1
x=y^3
x=1^3 x=-1^3
x=1 x=-1

Answer 16

They both work if you plug them into the equation

Answer 17

(2y +1)(y -1) =0
2y+1=0 or y-1=0

the idea is if you multiply two "things" and you get zero, then one or the other has to be zero

y-1=0
add +1 to both sides and you get
y= 1

now solve
2y + 1 = 0

the first step is add -1 to both sides. can you finish ?

Answer 18

2y= -1

now divide both sides by 2

Answer 19

y=-1/2
x=-0.125

Answer 20

so your two solutions are x=1 and x= -⅛

notice if we use x= -1 in the original equation
then the cube root of x is -1 (because -1*-1*-1= -1)
x^(⅓) = -1
and if we square it, i.e. x^(⅔) = -1*-1= 1
use those numbers in
2 x^(⅔) - x^(⅓) - 1 to get
2* 1 - (-1) - 1
2 + 1 - 1
2
we do not get zero... x=-1 is not a solution

Answer 21

if x = -⅛
what is the cube root of x ?

Answer 22

Ah okay

Answer 23

0.5?

Answer 24

½ * ½*½= ⅛
we want -⅛

Answer 25

notice -½ * -½ * -½ works (= -⅛)
so -½ is the cube root of -⅛
and if we square it to find x^(⅔) we get +¼

Answer 26

-1/2

Answer 27

now use x^⅔ = ¼ and x^⅓ = -½
in the original

Answer 28

You get. 0

Answer 29

yes

Answer 30

Mind helping me with one more similar problem? I already got it started

Answer 31

please make it a new post.

Answer 32

Okay.