Question

Sa se afle Nr nat x:. a)3•3²•3³•.....•3¹⁵=3⁵ ×. b)2•2²•2³•.....•2⁴³=4²×+¹. c)5•5²•5³•...•5¹⁹=25³×+². d)3•3²•3³•....3¹⁷=27×. e) 2•2³•2⁵•....•2³¹=2ײ. (URGENT!!!)VĂ ROG CINEVAAAAA!!!!!RPDDD!​

Answer 1

Răspuns:

a)3×3²×3³×....×3¹⁵=3^1+2+3+...+15=3^15×16÷2=3¹²⁰=3⁵×3¹¹⁵

b)2×2²×2³×...2⁴³=2^1+2+3+...+43=2^43×44÷2=2⁹⁴⁶=(2²)⁴⁷³=4⁴⁷³=4²×4⁴⁷¹

Answer 2

Răspuns:

Explicație pas cu pas:

Sa se afle Nr nat x:.

a)3•3²•3³•.....•3¹⁵=3⁵ × ⇔3⁽¹⁺²⁺³⁺⁴⁺⁵⁺   ⁺¹⁵⁾=3⁵ˣ  

Se foloseste formula sumei lui Gauss: (1+2+...+15)=15·16/2=15·8=120

⇔ 3¹²⁰=3⁵ˣ  ⇒120=5x  ⇒x=24

b)2•2²•2³•.....•2⁴³=4²×+¹ ⇔2⁽¹⁺²⁺³⁺⁴⁺⁵⁺   ⁺⁴³⁾=(2²)⁽²ˣ⁺¹⁾  

Se foloseste formula sumei lui Gauss:(1+2+...+43)=43·44/2=946

⇔2⁹⁴⁶=2⁽⁴ˣ⁺²⁾  ⇒946=4x+2  ⇒944=4x  ⇒x=236

c)5•5²•5³•...•5¹⁹=25³×+² ⇔5⁽¹⁺²⁺³⁺⁴⁺⁵⁺   ⁺¹⁹⁾ =(5²)⁽³ˣ⁺²⁾  

Se foloseste formula sumei lui Gauss: (1+2+...+19)=19·20/2=190

⇔5¹⁹⁰=5⁽⁶ˣ⁺⁴⁾  ⇒190=6x+4  ⇒186=6x  ⇒x=31

d)3•3²•3³•....3¹⁷=27× ⇔3⁽¹⁺²⁺³⁺⁴⁺⁵⁺   ⁺¹⁷⁾=(3³)ˣ  

Se foloseste formula sumei lui Gauss: (1+2+...+17)=17·18/2=153

⇔3¹⁵³=3³ˣ  ⇒153=3x  ⇒x=51

e) 2•2³•2⁵•....•2³¹=2ײ ⇔2⁽¹⁺³⁺⁵⁺    ⁺³¹⁾=2ײ

Se foloseste formula sumei lui Gauss: (1+3+5+...+31)=(1+2+3+...31)-(2+4+...+30)=31·32/2-2(1+2+...+15)=31·16-2·15·16/2=31·16-15·16=

16(31-15)=16·16=256=16²

⇔2⁽¹⁺³⁺⁵⁺   ⁺³¹⁾=2ײ  ⇔2²⁵⁶=2ײ  ⇒256=x²  ⇒16²=x²  ⇒x=16