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. x3y3 − x3y3 + (x+y)- (x−y) − x3y3 − 2xy+xy = xA xyxy + (x+y−xy)( x+y+/xy) xy(x2−x+y2= xy+ xy+x2 xy−xyy2= xxy−xyxy+ y2xy− (xx+6−x+2+3y−( xy)= (x−6xThe6xy)
2. 1−3a+5x−3+ (x−3âThisc∑this − d ∑e
3. 2(xThis+ay8+ThisThisx), −
4. x64 x + b−a,

Esercizi sui radicali

di Mattia Puddu

Semplifica i seguenti radicali, supponi positivi i radicandi letterali.

1. \( \sqrt{x^3y} - \sqrt{x^3y^3} + \sqrt{(x+y)^2-(x-y)^2} - \sqrt{x^3y-2x^2y} + xy = \) \( x\sqrt{xy}-xy\sqrt{xy} + \sqrt{(x+y+x-y)(x+y-x+y)} - \sqrt{xy(x^2-2x+1)} = \) \( x\sqrt{xy}-xy\sqrt{xy} + \sqrt{2x2y} - \sqrt{xy(x-1)^2} = x\sqrt{xy}-xy\sqrt{xy} + 2\sqrt{xy} - (x-1)\sqrt{xy} = \) \( (x-xy+2-x+1)\sqrt{xy} = (3-xy)\sqrt{xy}. \quad x≥1 \)
2. \( \sqrt{1-3a+3a^2-a^3} + \sqrt{1-2a+a^2-\sqrt{a^2-a^3-\sqrt{1-a}}} = \sqrt{-(a+1)^3} + \sqrt{(1-a)^2} = \) \( = \sqrt{a^2(1-a)} - \sqrt{1-a} = (1-a)\sqrt{1-a} + \sqrt{1-a} - |a|\sqrt{1-a} - \sqrt{1-a} = \) \( = (1-a+1-|a|-1)\sqrt{1-a} = -(a+1-|a|)\sqrt{1-a}. \quad a≤1 \)
3. \( \sqrt{2x^2+4xy+2y^2+\sqrt{3y^2+3x^2+6xy-\sqrt{10xy+5x^2+5y^2}}} = \) \( = \sqrt{2(x+y)^2} + \sqrt{3(x+y)^2 - \sqrt{5(x+y)^2}} = |x+y|\sqrt{2} + |x+y|\sqrt{3} - |x+y|\sqrt{5} = \) \( = |x+y|(\sqrt{2}+\sqrt{3}+\sqrt{5}). \)
4. \( \sqrt[4]{\frac{a^2b^2-a^2b}{b-a}} + \frac{a^2b^2-a^3b}{b-1} + \frac{b}{\sqrt{a^2}-\sqrt{a^2b}} = \) \( = \sqrt[4]{\frac{a^2b(b-1)^4}{a}} + \frac{a^2(b-a)}{b-1} + \frac{b}{\sqrt{a^2}-\sqrt{a^2b}} = \) \( = \sqrt{a^4b^2} + \frac{b}{\sqrt{a^2}-\sqrt{a^2b}} = \) \( = \sqrt{a^2b} + \frac{b}{\sqrt{a^2}-\sqrt{a^2b}} = \) \( = \sqrt{\frac{b}{a^2}-\frac{\sqrt{b}}{|a|}^2}. \quad b>a;b>1 \)

5. ⁴√₁₆₂ – ³√₃₂ + 5³√₁₆ – ³√₅₄ + ³√₂₅₀ =

= 3√₂ – 2√₂ + 10√₂ – 3√₂ + 5√₂ = √₂ + 12√₂.

6. (4 + √₂)² – (2√₂ – 1)² – 3(4√₂ + 2) =

= 16 + 2 + 8√₂ – 8 – 1 + 4√₂ – 12√₂ – 6 = 3.

7. (2√₅+1)(2√₅–1)–(√₅–1)²–(√₅–4)² =

= ^{20–1–5–1+2√5–5–16+8√5}⁄₂ = ^–8+10√5⁄₂ = –4 + 5√5.

8. 6√ab – 3√a – 7√ab + 2√a + 9√b + √a =

= –√ab + 9√b.

9. ⁸√₁₀a²b √3ab =

= ¹⁰√_2a^35a4b2a5b5⁄_2a = ¹⁰√₂^243a6b7⁄₂.

10. √^3ab2⁄_9b² √^c⁄_c ^a⁄₃ =

= √^3ab2⁄_c ^c⁄_9b² √^a⁄₃ ^a2⁄₉ ^a⁄₃.

11. √^32x2⁄_3y √^6xy2⁄₃ =

= √^64x6⁄_9y² √^6y3⁄_x³ √^6xy2⁄₃ √^4x2y3⁄₂₇.