a) \(2^{135}=2^{3.45}=\left(2^3\right)^{45}=8^{45}\)
\(3^{90}=3^{2.45}=\left(3^2\right)^{45}=9^{45}\)
Vì \(8^{45}< 9^{45}\)nên \(2^{135}< 3^{90}\)

b) \(4^{75}=4^{3.25}=\left(4^3\right)^{25}=64^{25}\)
\(3^{100}=3^{4.25}=\left(3^4\right)^{25}=81^{25}\)
Vì \(64^{25}< 81^{25}\)nên \(4^{75}< 3^{100}\)

c) \(4^{100}=4^{4.25}=\left(4^4\right)^{25}=256^{25}\)
\(9^{75}=9^{3.25}=\left(9^3\right)^{25}=729^{25}\)
Vì \(256^{25}< 729^{25}\)nên \(^{4^{100}< 9^{75}}\)

a, 24^10 < 3^30 + 4^30 + 5^30

b, 2^100 < 5^50 < 3^75.

\(a=2^{100}=\left(2^4\right)^{25}=16^{25}\)

\(b=3^{75}=\left(3^3\right)^{25}=27^{25}\)

\(c=5^{50}=\left(5^2\right)^{25}=25^{25}\)

Vì \(16^{25}< 25^{25}< 27^{25}\)

\(\Rightarrow a< c< b\)

\(a=2^{100},b=3^{75},c=5^{50}\\ \Rightarrow a=30^{85},b=30^{65},c=30^{44}\\ \Rightarrow a>b>c\)

Ta có:

\(a=2^{100}=2^{4\cdot25}=\left(2^4\right)^{25}=16^{25}\)

\(b=3^{75}=3^{3\cdot25}=\left(3^3\right)^{25}=27^{25}\)

\(c=5^{50}=5^{2\cdot25}=\left(5^2\right)^{25}=25^{50}\)

Ta thấy: 

\(16< 25< 27\)

\(\Rightarrow16^{25}< 25^{25}< 27^{25}\)

\(\Rightarrow2^{100}< 5^{50}< 3^{75}\)

\(\Rightarrow a< c< b\)

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

\(3< 25=>3^{100}< 25^{100}=>3^{100}< 5^{200}\)

\(\frac{75^{20}}{45^{10}.25^{15}}=\frac{25^{20}.3^{20}}{3^{10}.3^{10}.5^{10}.25^{15}}=\frac{25^{20}}{25^5.25^{15}}=1\)

\(=>75^{20}=45^{10}.25^{15}\left(dpcm\right)\)

P/S:nếu a=b=>a:b=1 mk làm theo cách đó cho nhanh mà bn ghi sai đề r

6/8>5/7

9/15=3/5

24/36<75/100

\(\dfrac{6}{8}\) ....\(\dfrac{5}{7}\)

\(\dfrac{6}{8}=\dfrac{30}{40}\);  \(\dfrac{5}{7}\)  = \(\dfrac{30}{42}\)

Vì  \(\dfrac{30}{40}\)  > \(\dfrac{30}{42}\) nên \(\dfrac{6}{8}\) > \(\dfrac{5}{7}\)

\(\dfrac{9}{15}\) ....\(\dfrac{3}{5}\)

\(\dfrac{9}{15}\) = \(\dfrac{9:3}{15:3}\) = \(\dfrac{3}{5}\)

Vậy \(\dfrac{9}{15}\) = \(\dfrac{3}{5}\)

\(\dfrac{24}{36}\) ....\(\dfrac{75}{100}\)

\(\dfrac{24}{36}\) = \(\dfrac{24:12}{36:12}\) = \(\dfrac{2}{3}\) = 1 - \(\dfrac{1}{3}\);              \(\dfrac{75}{100}\) = \(\dfrac{75:25}{100:25}\) = \(\dfrac{3}{4}\) = 1 - \(\dfrac{1}{4}\)

vì \(\dfrac{1}{3}\) > \(\dfrac{1}{4}\) nên     1 - \(\dfrac{1}{3}\) < 1 - \(\dfrac{1}{4}\)

Vậy \(\dfrac{24}{36}\) < \(\dfrac{75}{100}\)

6/8>5/7

9/15=3/5

24/36<75/100

❤❤❤

\(2^{100}=\left(2^4\right)^{25}=16^{25};3^{75}=\left(3^3\right)^{25}=27^{25}\)

\(\Leftrightarrow2^{100}

Ta có :

\(2^{100}=\left(2^4\right)^{25}=16^{25}\)

\(3^{75}=\left(3^3\right)^{25}=27^{25}\)

\(5^{50}=\left(5^2\right)^{25}=25^{25}\)

Do \(16^{25}< 25^{25}< 27^{25}\)

\(\Rightarrow2^{100}< 5^{50}< 3^{75}\)

Áp dụng a/b > 1 => a/b > a+m/b+m (a,b,m thuộc N*)

=> \(\frac{5^{100}+6}{5^{100}+4}>\frac{5^{100}+6+1}{5^{100}+4+1}\)

                       \(>\frac{5^{100}+7}{5^{100}+5}\)