Rationalize this expression:
2 on root 3, minus root 2 on 3
To rationalize the given expression, we need to eliminate the square roots in the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
The given expression is: 2/(√3) - (√2)/3
To eliminate the square root in the denominator, we multiply the numerator and denominator of the first term by (√3), and the numerator and denominator of the second term by 3:
(2/(√3)) * (√3)/(√3) - (√2)/3 * 3/3
= (2√3)/(√9) - (3√2)/9
Simplifying further, we have:
2√3/3 - 3√2/9
To combine the terms, we need to find a common denominator. The least common multiple (LCM) of 3 and 9 is 9.
Therefore, multiplying the first term by 3/3 and the second term by 1/1, we get:
(2√3 * 3)/(3 * 3) - (3√2 * 1)/(9 * 1)
= 6√3/9 - 3√2/9
Now that the terms have the same denominator, we can simply subtract them:
(6√3 - 3√2)/9
Thus, the simplified expression after rationalization is (6√3 - 3√2)/9.
To rationalize the expression (2√3 - √2) / 3, you can multiply both the numerator and the denominator by the conjugate of the numerator, which is (2√3 + √2):
[(2√3 - √2) / 3] * [(2√3 + √2) / (2√3 + √2)]
Expanding the numerator and the denominator, we have:
[(2√3 * 2√3) + (2√3 * √2) - (√2 * 2√3) - (√2 * √2)] / [3 * (2√3 + √2)]
Simplifying, we get:
[(4 * 3) + (2√6) - (2√6) - 2] / (6√3 + 3√2)
Simplifying further, we have:
[12 - 2] / (6√3 + 3√2)
Finally:
10 / (6√3 + 3√2)