Question

(2¹•2²•2³ • ... • 2⁵⁰): (2¹⁷)⁷⁵
(3¹•3²•3³ • ... • 3⁸⁰): (3¹•3³•3⁵• ... • 3¹¹¹)
(5¹•5³•5⁵• ... • 5⁹⁹): (5⁵•5¹⁰•5¹⁵• ... •5¹⁵⁰)
(7¹•7⁶•7¹¹• ... • 7³⁰¹): (7¹•7⁴•7⁷•7¹⁰• ... •7¹⁵⁴)
(4¹•4²•4³• ... • 4¹⁵⁰): (2¹•2³•2⁵• ... • 2²⁵¹)
(9²•9⁴•9⁶• ... • 9⁸⁰): (3¹•3²•3³• ... •3⁵⁰)
Va rog frumos!!! URGENTTTT!! DAU COROANA, PROMIT!!​

Answer 1

Explicație pas cu pas:

a)

(2¹•2²•2³ • ... • 2⁵⁰) : (2¹⁷)⁷⁵ =

$= {2}^{1 + 2 + 3 + ... + 50} : {2}^{17 \cdot 75} = {2}^{ \frac{50 \cdot 51}{2} } : {2}^{1275}$

$= {2}^{1275} : {2}^{1275} = \bf 1$

b)

(3¹•3²•3³ • ... • 3⁸⁰) : (3¹•3³•3⁵• ... • 3¹¹¹) =

$= {3}^{1 + 2 + 3 + ... + 80} : {3}^{1 + 3 + 5 + ... + 111}$

$= {3}^{ \frac{80 \cdot 81}{2} } : {3}^{1 + 3 + 5 + ... + (2 \cdot 56 - 1)}$

$= {3}^{3240} : {3}^{ ({56}^{2}) } = {3}^{3240} : {3}^{3136}$

$= {3}^{3240 - 3136} = \bf {3}^{104}$

c)

(5¹•5³•5⁵• ... • 5⁹⁹) : (5⁵•5¹⁰•5¹⁵• ... •5¹⁵⁰) =

$= {5}^{1 + 3 + 5 + ... + 99} : {5}^{5 + 10 + 15 + ... + 150}$

$= {5}^{1 + 3 + 5 + ... + (2 \cdot 50 - 1)} : {5}^{5 \cdot (1 + 2 + 3 + ... + 30)}$

$= {5}^{ ({50}^{2}) } : {5}^{5 \cdot \frac{30 \cdot 31}{2} } = {5}^{2500} : {5}^{2325}$

$= {5}^{2500 - 2325} = \bf {5}^{175}$

d)

(7¹•7⁶•7¹¹• ... • 7³⁰¹) : (7¹•7⁴•7⁷•7¹⁰• ... •7¹⁵⁴) =

$= {7}^{1 + 6 + 11 + ... + 301} : {7}^{1 + 4 + 7 + 10 + ... + 154}$

$= {7}^{ \frac{(1 + 301) \cdot 61}{2} } : {7}^{\frac{(1 + 154) \cdot 52}{2}} = {7}^{ \frac{302 \cdot 61}{2} } : {7}^{\frac{155 \cdot 52}{2}}$

$= {7}^{9211} : {7}^{4030} = {7}^{9211 - 4030} = \bf {7}^{5181}$

e)

(4¹•4²•4³• ... • 4¹⁵⁰) : (2¹•2³•2⁵• ... • 2²⁵¹) =

$= {4}^{1 + 2 + 3 + ... + 150} : {2}^{1 + 3 + 5 + ... + 251}$

$= {4}^{ \frac{150 \cdot 151}{2} } : {2}^{1 + 3 + 5 + ... + (2 \cdot 126 - 1)}$

$= {4}^{11325} : {2}^{( {126}^{2} )} = {( {2}^{2} )}^{11325} : {2}^{15876}$

$= {2}^{22650} : {2}^{15876} = {2}^{22650 - 15876} = \bf {2}^{6774}$

f)

(9²•9⁴•9⁶• ... • 9⁸⁰) : (3¹•3²•3³• ... •3⁵⁰) =

$= {9}^{2 + 4 + 6 + ... + 80} : {3}^{1 + 2 + 3 + ... + 50}$

$= {( {3}^{2}) }^{2 \cdot (1 + 2 + 3 + ... + 40)} : {3}^{ \frac{50 \cdot 51}{2} }$

$= {3}^{4 \cdot \frac{40 \cdot 41}{2} } : {3}^{1275} = {3}^{3280} : {3}^{1275}$

$= {3}^{3280 - 1275} = \bf {3}^{2005}$