How can you write the expression with a rationalized denominator?
^3sqrt2/^3sqrt6
To rationalize the denominator of the expression ^3sqrt2/^3sqrt6, multiply both the numerator and denominator by ^3sqrt2, which is the conjugate of the denominator:
(^3sqrt2/^3sqrt6) * (^3sqrt2/^3sqrt2)
= (^3sqrt(2*2))/(^3sqrt(6*2))
= (^3sqrt4)/(^3sqrt12)
= ^3sqrt4/^3(sqrt12)
Therefore, the rationalized expression is ^3sqrt4/^3sqrt12.
Are you sure?
Apologies for the mistake in my previous response. Let's revisit the rationalization process for the expression ^3sqrt2/^3sqrt6.
Given expression: ^3sqrt2/^3sqrt6
To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator, which in this case is ^3sqrt(6).
^3sqrt2/^3sqrt6 * ^3sqrt(6)/^3sqrt(6)
= ^3sqrt(2*6)/(^3sqrt6 * ^3sqrt6)
= ^3sqrt12/^3(6)
= ^3sqrt12/^3(2*3)
= ^3sqrt12/^3^2
Therefore, the rationalized expression is ^3sqrt12/^3^2.
I don't think that's right.
Apologies for the confusion in my previous responses. Let's correct the rationalization process for the expression ^3sqrt2 / ^3sqrt6.
Given expression: ^3sqrt2 / ^3sqrt6
To rationalize the denominator, we multiply both the numerator and the denominator by ^3sqrt6 (the conjugate of the denominator):
^3sqrt2 / ^3sqrt6 * ^3sqrt6 / ^3sqrt6
= (^3sqrt2 * ^3sqrt6) / (^3sqrt6 * ^3sqrt6)
= ^3sqrt(2*6) / (^3sqrt6)^2
= ^3sqrt12 / 6
= ^3sqrt12 / 6
Therefore, the expression with a rationalized denominator is ^3sqrt12 / 6.
I don't think that's right either.
I apologize for the errors in my previous responses. Let's correct the rationalization process for the expression ^3sqrt2 / ^3sqrt6.
Given expression: ^3sqrt2 / ^3sqrt6
To rationalize the denominator, we multiply both the numerator and the denominator by ^3sqrt(6^2), which is ^3sqrt36 since 6^2 equals 36.
^3sqrt2 / ^3sqrt6 * ^3sqrt36 / ^3sqrt36
= ^3sqrt(2*36) / ^3sqrt(6*36)
= ^3sqrt72 / ^3sqrt216
Now simplify by finding perfect cube factors:
^3sqrt72 = ^3sqrt(8 * 9) = ^3sqrt8 * ^3sqrt9 = 2 * ^3sqrt9 = 2 * 3 = 6
^3sqrt216 = ^3sqrt(72 * 3) = ^3sqrt72 * ^3sqrt3 = 6 * ^3sqrt3
Therefore, the rationalized expression is 6 / (6 * ^3sqrt3) or simply 1 / ^3sqrt3.
Please give me a correct response...
Apologies for the errors in my previous responses. Let's correct the rationalization process for the expression ^3sqrt2 / ^3sqrt6.
Given expression: ^3sqrt2 / ^3sqrt6
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is ^3sqrt6:
^3sqrt2 / ^3sqrt6 * ^3sqrt6 / ^3sqrt6
= (^3sqrt2 * ^3sqrt6) / (^3sqrt6 * ^3sqrt6)
= ^3sqrt(2*6) / 6
= ^3sqrt12 / 6
Since ^3sqrt12 can be simplified as ^3sqrt(4*3) = ^3sqrt4 * ^3sqrt3 = 2 * ^3sqrt3 = 2^3sqrt3, the rationalized expression is:
2^3sqrt3 / 6
Now simplify this expression by dividing both numerator and denominator by 2:
= 2^3sqrt3 / 6 = ^3sqrt3 / 3
Therefore, the expression ^3sqrt2 / ^3sqrt6 with a rationalized denominator is ^3sqrt3 / 3.