Big Question:
To rationalize a fraction means to remove all non-rational numbers from the denominator of the fraction. Rationalise (a^2)/(3+√b^3)
by multiplying the numerator and
denominator by 3 − √�b^3, and then evaluate if b = a^2 and a = 2. Show all of your working.
so, did you do what they asked? What do you get?
Sorry for late reply, but after multiplying 3 − √�b^3 in num and dom, I am stuck at a^(6/2)/b^(3/2), which I believe, is incorrect.
To rationalize the given fraction (a^2)/(3+√b^3), we need to remove the square root (√) from the denominator.
1. Start by multiplying the numerator and denominator by the conjugate of the denominator, which is 3 - √b^3. This will help eliminate the square root.
(a^2) * (3 - √b^3)
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(3 + √b^3) * (3 - √b^3)
2. Now, let's simplify the denominator. When we multiply the conjugate, it eliminates the square root term, following the pattern (a + b)(a - b) = a^2 - b^2.
= (a^2 * (3 - √b^3)) / ((3)^2 - (√b^3)^2)
= (a^2 * (3 - √b^3)) / (9 - b^3)
3. Since we have b = a^2, we can substitute it into the expression.
= (a^2 * (3 - √(a^2)^3)) / (9 - (a^2)^3)
= (a^2 * (3 - √a^6)) / (9 - a^6)
4. Given a = 2, replace a with 2 in the expression.
= (2^2 * (3 - √(2^2)^6)) / (9 - (2^2)^6)
= (4 * (3 - √64)) / (9 - 64)
= (4 * (3 - 8)) / (9 - 64)
= (4 * (-5)) / (-55)
= -20 / -55
5. Finally, simplify the fraction.
= 4/11
Therefore, the rationalized form of the fraction (a^2)/(3+√b^3), when b = a^2 and a = 2, is 4/11.