How can you write the expression with a rationalized denominator?
^3 sqrt 2 / ^3 sqrt 4
a.3sqrt32/2
b. 2+^3sqrt4/4
c. ^3sqrt4/4
d. ^3sqrt4/2
First, we can simplify the expression ^3√4 to 2. Then, we can rewrite the original expression as ^3√2/2. To rationalize the denominator, we can multiply both the numerator and denominator by ^3√4, which gives us ^3√2 * ^3√4 / (2 * ^3√4). Simplifying further, we get ^3√(2*4) / (2* ^3√(4*4)) or ^3√8 / (2*4), which simplifies to ^3√2 / 4. Therefore, the answer is (c) ^3√4/4.
To rationalize the denominator, we need to eliminate the cube root from the denominator.
The expression ^3 sqrt 2 / ^3 sqrt 4 can be simplified as follows:
^3 sqrt 2 / ^3 sqrt 4
Since the denominator is ^3 sqrt 4, we can rewrite it as ^(3/2) sqrt 2.
Now, let's simplify the numerator by multiplying the cube root of 2 by the cube root of 4 to get:
^3 sqrt 2 * ^(3/2) sqrt 2
Since the bases are the same (^3 sqrt 2 and ^(3/2) sqrt 2), we can combine the exponents:
^3 sqrt(2 * 2) = ^3 sqrt 4
So, the expression becomes:
^3 sqrt 4 / ^(3/2) sqrt 4
Since ^3 sqrt 4 is the same as ^3 sqrt(2^2), it simplifies to 2.
And ^(3/2) sqrt 4 is the same as ^(3/2) sqrt(2^2), which simplifies to just 2.
Therefore, the expression simplifies to:
2 / 2
And this can be further simplified to:
1
Therefore, the answer is (d) ^3 sqrt 4 / 2.