\(3^{200}=9^{100}>4^{100}\\ 5^{200}=25^{100}< 64^{100}=4^{300}\\ 6^{50}=36^{25}>7^{25}\\ 8^{40}=64^{20}>10^{20}\\ 16^{20}=256^{10}>32^{10}\)

tick mik nha!!

3200=9100>41005200=25100<64100=4300650=3625>725840=6420>10201620=25610>3210

a: \(3^{200}=\left(3^2\right)^{100}=9^{100}>4^{100}\)

b: \(5^{200}=\left(5^2\right)^{100}=25^{100}\)

\(4^{300}=\left(4^3\right)^{100}=64^{100}\)

mà 25<64

nên \(5^{200}< 4^{300}\)

c: \(6^{50}=\left(6^2\right)^{25}=36^{25}>7^{25}\)

a) \(31^{11}< 32^{11}=\left(2^5\right)^{11}=2^{55}\)

\(\Rightarrow\)\(31^{11}< 2^{55}\)

\(17^{14}>16^{14}=\left(2^4\right)^{14}=2^{56}\)

\(\Rightarrow\)\(17^{16}>2^{56}\)

Mà \(2^{55}< 2^{56}\)

\(\Rightarrow\)\(31^{11}< 17^{14}\)

b và c chứng minh tương tự 

\(3\sqrt{3}-2\sqrt{2}=\sqrt{27}-\sqrt{8}>\sqrt{25}-\sqrt{9}=5-3\)

\(\Rightarrow3\sqrt{3}-2\sqrt{2}>2\)

ta có:

\(199^{20}=\left(199^4\right)^5=1568239201^5\)

\(2003^{15}=\left(2003^3\right)^5=8036054027^5\)

vì \(1568239201>1\)

    \(8036054027>1\)

\(1568239201< 8036054027\)

=>\(199^{20}< 2003^{15}\)

199²⁰=(199⁴)⁵=1568239201⁵

2003¹⁵=(2003³)⁵=8036054027⁵

Vì 1568239201<8036054027

Vậy 199²⁰ <2003¹⁵

\(199^{20}< 2003^{15}\)

199^20<2003^15

199²⁰=(199⁴)⁵

2003¹⁵=(2003³)⁵

Vì 199⁴<2003³=>199²⁰<2003¹⁵

Lũy thừa để so sánh là :

199²⁰ = ( 199⁴⁵ )

2003¹⁵ = ( 2003³⁵ )

Do : 199⁴ < 2003³ => 199²⁰ < 2003¹⁵ .

Ta có: 199^20<200^20 và 2000^15<2003^15

200^20=200^(4.5)=(200^4)^5

2000^15=2000^(3.5)=(2000^3)^5

So sánh: 200^4 và 2000^3

200^4=200^3.200

2000^3=(200.10)^3=200^3.10^3=200^3.1000

Vì 200<1000 =>200^4<2000^3 =>200^20<2000^15 =>199^20<2003^15.

Câu hỏi này khó wá ko thể trả lời đc .

{'-'}...{'^'}

199\(^{20}\)<200\(^{20}\) =(2\(^3\).5\(^2\))\(^{20}\)=2\(^{60}\).5\(^{40}\)

2003\(^{15}\)>2000\(^{15}\)=(2.10\(^3\))\(^{15}\)=(2\(^4\).5\(^3\))\(^{15}\)=2\(^{60}\).5\(^{45}\)

\(\Rightarrow\)2\(^{60}\).5\(^{40}\)<2\(^{60}\).5\(^{45}\)\(\Rightarrow\)199\(^{20}\)<2003\(^{15}\)

nghuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu

ta có 199^20 < 200^20 = 2^20 x 10^40

2003^15 >2000^15 = 2^15 x 10^45 mặt khác để thấy được 2^20 x 10^40<2^15 x 10^45

suy ra: 2003^15 > 199^20