Question

Scrieți sub forma unei singure puteri:
a) 1,4×1,4^2×1,4^3×. ×1,4^99
b) 2,5^2×2,5^4×2,5^6×. ×2,5^100
c) 2,25×2,25^3×2,25^5×. ×2,25^101
d) {[(1,7^2)^3]^5}^7×1,7^1000​

Va rog,este urgent!.

Answer 1

Explicație pas cu pas:

a) 1,4×1,4^2×1,4^3×. ×1,4^99

$1,4 \times 1,4^{2} \times 1,4^{3} \times ... \times 1,4^{99} = \\ = {1.4}^{1 + 2 + 3 + ... + 99} = {1.4}^{ \frac{99 \times 100}{2} } = {1.4}^{4950}$

b) 2,5^2×2,5^4×2,5^6×. ×2,5^100

$2,5^{2} \times 2,5^{4} \times 2,5^{6} \times ... \times 2,5^{100} = \\ = {2.5}^{2 + 4 + 6 + ... + 100} = {2.5}^{2(1 + 2 + 3 + ... + 50)} \\ = {2.5}^{2 \times \frac{50 \times 51}{2} } = {2.5}^{50 \times 51} = {2.5}^{2550}$

c) 2,25×2,25^3×2,25^5×. ×2,25^101

$1 + 3 + 5 + ... + 101 = \\ = 1 + (1 + 2) + (1 + 4) + ... + (1 + 100) \\ = (1 + 1 + ... + 1) + 2 + 4 + ... + 100 \\ = 101 \times 1 + 2(1 + 2 + ... + 50) \\ = 101 + 2 \times \frac{50 \times 51}{2} = 101 + 50 \times 51 = 2651$

=>

$2,25 \times 2,25^{3} \times 2,25^{5} \times ... \times 2,25^{101} = \\ = {2.25}^{1 + 3 + 5 + ... + 101} = {2.25}^{2651} \\ = { \left({(1.5)}^{2} \right)}^{2651} = {1.5}^{2 \times 2651} = {1.5}^{5302}$

d) {[(1,7^2)^3]^5}^7×1,7^1000

$\{\left[ \left(1,7^{2} \right)^{3} \right]^{5} \}^{7} \times 1,7^{1000} = {1.7}^{2 \times 3 \times 5 \times 7} \times {1.7}^{1000} \\ = {1.7}^{210 + 1000} = {1.7}^{1210}$