Esercizi sui radicali
di Mattia Puddu
Semplifica i seguenti radicali, supponi positivi i radicandi letterali.
1. cuberoot(x/y3 - 1/y2) + cuberoot(xy3 - y4) - cuberoot(8x - 8y) =
= cuberoot((x-y)/y3) + cuberoot(y3(x-y)) - cuberoot(8(x-y)) = cuberoot(x-y)/y + y cuberoot(x-y) - 2 cuberoot(x-y)
= (cuberoot(x-y) + y2 cuberoot(x-y) - 2y cuberoot(x-y))/y = (1+y2-2y)/y cuberoot(x-y) = (y-1)2/y cuberoot(x-y).
2. fourthroot((a2-b2)/(x4-y4)) fourthroot((a-b)3/(x-y)3) fourthroot((x+y)/(a-2b)) =
= fourthroot(((a+b)(a-b)(a-b)3(x+y))/((x2+y2)(x2-y2)(x-y)3(a-2b))) = fourthroot(((a+b)(a-b)(a-b)3(x+y))/((x2+y2)(x+y)(x-y)(x-y)3(a-2b)))
= fourthroot(((a+b)(a-b)4)/((x2+y2)(x-y)4(a-2b))) = (a-b)/(x-y) fourthroot((a+b)/((x2+y2)(a-2b))).
3. (sqrt((a2+2ab+b2)/(a3-3a2b+3ab2-b3)) cuberoot((a2-b(a-b))/(a+b)))/sqrt((a3+b3)/(a2-b2))
= sqrt(((a+b)2/(a-b)3) * ((a2-ab+b2)/(a+b)) * (((a+b)(a-b))/((a+b)(a2-ab+b2)))) = sqrt((a+b)/(a-b)2) = sqrt(a+b)/(a-b).
4. sqrt((a3+2a2+a)/(a2+6a+9)) + sqrt((a3+4a2+4a)/(a2+6a+9)) - sqrt(a3/(a2+6a+9))
= sqrt(a(a+1)2/(a+3)2) + sqrt(a(a+2)2/(a+3)2) - a sqrt(a)/(a+3) = (a+1)/(a+3) sqrt(a) + (a+2)/(a+3) sqrt(a) - a/(a+3) sqrt(a)
= (a+1+a+2-a)/(a+3) sqrt(a) = (a+3)/(a+3) sqrt(a) = sqrt(a).
5. sqrt((a2-1)/(a2+a-2)) / cuberoot((a2-4)/(a+1)) sixthroot((a+2)/(a2+2a+1))
= sixthroot(((a+1)3(a-1)3)/((a+2)3(a-1)3) * ((a+1)2)/((a+2)2(a-2)2) * (a+2)/(a+1)2) = sixthroot((a+1)3/((a+2)4(a-2)2)).
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