Semplifica i seguenti radicali, supponi positivi i radicandi letterali

Esercizi sui radicali

di Mattia Puddu

Semplifica i seguenti radicali, supponi positivi i radicandi letterali.

1. √(x³y) - √(x³y³) + √((x + y)² - (x - y)²) - √(x³y - 2x²y + xy) =

x√xy - xy√xy + √((x + y + x - y)(x + y - x + y)) - √(xy(x² - 2x + 1)) =

x√xy - xy√xy + √(2x²y) - √(xy(x - 1)²) = x√xy - xy√xy + 2√xy - (x - 1)√xy =

(x - xy + 2 - x + 1)√xy = (3 - xy)√xy.     x ≥ 1

2. √(1 - 3a + 3a² - a³) + ⁴√(1 - 2a + a²) - √(a² - a³) - √(1 - a) = √((-a + 1)³) + ⁴√((1 - a)²) =

= √(a²(1 - a)) - √(1 - a) = (1 - a)√(1 - a) + √(1 - a) - |a|√(1 - a) - √(1 - a) =

= (1 - a + 1 - |a| - 1)√(1 - a) = (-a + 1 - |a|)√(1 - a).     a ≤ 1

3. √(2x² + 4xy + 2y²) + √(3y² + 3x² + 6xy) - √(10xy + 5x² + 5y²) =

= √(2(x + y)²) + √(3(x + y)²) - √(5(x + y)²) = |x + y|√2 + |x + y|√3 - |x + y|√5 =

= |x + y|(√2 + √3 + √5).

4. ⁴√((a²b² - a²b)/(b - a)) ⁴√((a²b² - a³b)/(b - 1)) + √(b/a²) - √(a²b) =

= ⁴√((a²b(b - 1))/(b - a)) ⁴√((a²b(b - a))/(b - 1)) + √(b/a²) - √(a²b) =

= ⁴√(a⁴b²) + √(b/a²) - √(a²b) =

= √(a²b) + √(b/a²) - √(a²b) =

= √(b/a²) = √b/|a|.     b > a; b > 1

5. ⁴sqrt(162) - ⁴sqrt(32) + ³sqrt(16) - ³sqrt(54) + ³sqrt(250) =

= 3⁴sqrt(2) - 2⁴sqrt(2) + 10³sqrt(2) - 3³sqrt(2) + 5³sqrt(2) = ⁴sqrt(2) + 12³sqrt(2).

6. (4 + sqrt(2))² - (2sqrt(2) - 1)² - 3(4sqrt(2) + 2) =

= 16 + 2 + 8sqrt(2) - 8 - 1 + 4sqrt(2) - 12sqrt(2) - 6 = 3.

7. ((2sqrt(5) + 1)(2sqrt(5) - 1) - (sqrt(5) - 1)² - (sqrt(5) - 4)²)/2 =

= (20 - 1 - 5 - 1 + 2sqrt(5) - 5 - 16 + 8sqrt(5))/2 = (-8 + 10sqrt(5))/2 = -4 + 5sqrt(5).

8. 6sqrt(ab) - 3sqrt(a) - 7sqrt(ab) + 2sqrt(a) + 9sqrt(b) + sqrt(a) =

= -sqrt(ab) + 9sqrt(b).

9. (⁵sqrt(a²b)/¹⁰sqrt(2a))sqrt(3ab) =

= ¹⁰sqrt((3⁵a⁴b²a⁵b⁵)/(2a)) = ¹⁰sqrt(243a⁸b⁷/2).

10. (sqrt(3ab²/c)/sqrt(9b²/c))sqrt(a/3) =

= sqrt(3ab²/c)sqrt(c/(9b²))sqrt(a/3) = sqrt(a²/9) = a/3.

11. ((³sqrt(2x²/(3y)))/sqrt(x/y))⁶sqrt(xy²/3) =

= ⁶sqrt((4x⁴/(9y²))(y³/x³)(xy²/3)) = ⁶sqrt(4x²y³/27).

12. sqrt(sqrt((2x+3)³)) / sixthroot(2x+3) =

= fourthroot((2x+3)³) / sixthroot(2x+3) = twelfthroot(((2x+3)⁹)/((2x+3)²)) = twelfthroot((2x+3)⁷).

13. 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 + y³cuberoot(x-y) - 2cuberoot(x-y)

= (cuberoot(x-y) + y²cuberoot(x-y) - 2y cuberoot(x-y))/y = (1 + y² - 2y)/y cuberoot(x-y) =

= (y-1)²/y cuberoot(x-y).

14. 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))).

15. [sqrt((a²+2ab+b²)/(a³-3a²b+3ab²-b³)) sqrt((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).

16. 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)²) = (a+1)/(a+3) sqrt(a) + (a+2)/(a+3) sqrt(a) - a/(a+3) sqrt(a) = sqrt(a).

17.

18. 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)).