Question

Exercițiul 5 punctele a și b repede

Answer 1

$a)( {3}^{1 + 2 + 3 + ... + 100 } + 2 \times {3}^{5050} ) \div {3}^{5051}$

$= ( {3}^{ \frac{100(100 + 1)}{2} } + 2 \times {3}^{5050} ) \div {3}^{5051}$

$= ( {3}^{ \frac{100 \times 101}{2} } + 2 \times {3}^{5050} ) \div {3}^{5051}$

$= ( {3}^{50 \times 101} + 2 \times {3}^{5050} ) \div {3}^{5051}$

$= ( {3}^{5050} + 2 \times {3}^{5050} ) \div {3}^{5051}$

$= {3}^{5050} (1 + 2) \div {3}^{5051}$

$= {3}^{5050} \times 3 \div {3}^{5051}$

$= {3}^{5050 + 1 - 5051}$

$= {3}^{5051 - 5051}$

$= {3}^{0}$

$= 1$

$b)( {5}^{2 + 4 + 6 + ... + 200} + 4 \times {5}^{10100} ) \div {5}^{10100}$

$= ( {5}^{2(1 + 2 + 3 + ... + 100)} + 4 \times {5}^{10100} ) \div {5}^{10100}$

$= ( {5}^{2 \times \frac{100(100 + 1)}{2} } + 4 \times {5}^{10100} ) \div {5}^{10100}$

$= ( {5}^{100(100 + 1)} + 4 \times {5}^{10100} ) \div {5}^{10100}$

$= ( {5}^{100 \times 101} + 4 \times {5}^{10100} ) \div {5}^{10100}$

$= ( {5}^{10100} + 4 \times {5}^{10100} ) \div {5}^{10100}$

$= {5}^{10100} (1 + 4) \div {5}^{10100}$

$= {5}^{10100} \times 5 \div {5}^{10100}$

$= {5}^{10100 + 1 - 10100}$

$= {5}^{10101 - 10100}$

$= {5}^{1}$

$= 5$