a) Ta có: \(25^{15}=\left(5^2\right)^{15}=5^{30}\)

               \(8^{10}.3^{30}=\left(2^3\right)^{10}.3^{30}\)\(=2^{30}.3^{30}=6^{30}\)

Vì \(5^{30}< 6^{30}\)nên \(25^{15}< 8^{10}.3^{30}\)

b) Ta có: \(\frac{4^{15}}{7^{30}}=\frac{\left(2^2\right)^{15}}{7^{30}}=\frac{2^{30}}{7^{30}}\)

\(\frac{8^{10}.3^{30}}{7^{30}.4^{15}}=\frac{\left(2^3\right)^{10}.3^{30}}{7^{30}.\left(2^2\right)^{15}}=\frac{2^{30}.3^{30}}{7^{30}.2^{30}}=\frac{3^{30}}{7^{30}}\)

Vì \(2^{30}< 3^{30}\)nên \(\frac{2^{30}}{7^{30}}< \frac{3^{30}}{7^{30}}\)hay \(\frac{4^{15}}{7^{30}}< \frac{8^{10}.3^{30}}{7^{30}.4^{15}}\)

_Học tốt_

a) 25¹⁵ và 8¹⁰. 3³⁰

25¹⁵ = (5² ) ¹⁵ = 5³⁰

8¹⁰. 3³⁰ = (2³ )¹⁰. 3³⁰ = 2³⁰. 3³⁰ = 6³⁰

Vì 5³⁰< 6³⁰

nên 25¹⁵< 8¹⁰. 3³⁰

b) \(\frac{4^{15}}{7^{30}}\)và \(\frac{8^{10}.3^{30}}{7^{30}.4^{15}}\)

\(\frac{4^{15}}{7^{30}}=\frac{\left(2^2\right)^{15}}{7^{30}}=\frac{2^{30}}{7^{30}}\)

\(\frac{8^{10}.3^{30}}{7^{30}.4^{15}}=\frac{\left(2^3\right)^{10}.3^{30}}{7^{30}.\left(2^2\right)^{15}}=\frac{2^{30}.3^{30}}{7^{30}.2^{30}}=\frac{3^{30}}{7^{30}}\)

Vì \(\frac{2^{30}}{7^{30}}< \frac{3^{30}}{7^{30}}\)

nên \(\frac{4^{15}}{7^{30}}< \frac{8^{10}.3^{30}}{7^{30}.4^{15}}\)

a)\(25^{15}=5^{2^{15}}=5^{30}\)

\(8^{10}.3^{30}=2^{3^{10}}.3^{30}=\left(2.3\right)^{30}=6^{30}\)

\(5^{30}< 6^{30}=>25^{15}< 8^{10}.3^{30}\)

b)\(\frac{4^{15}}{7^{30}}=\frac{2^{2^{15}}}{7^{30}}=\frac{2^{30}}{7^{30}}=\left(\frac{2}{7}\right)^{30}\)

\(\frac{8^{10}.3^{30}}{7^{30}.4^{15}}=\frac{2^{30}.3^{30}}{7^{30}.2^{30}}=\frac{6^{30}}{14^{30}}=\left(\frac{6}{14}\right)^{30}=\left(\frac{3}{7}\right)^{30}\)

Vì hai số có mũ bằng 30 nên ta so sánh :\(\frac{2}{7}< \frac{3}{7}\)

=>\(\frac{4^{15}}{7^{30}}< \frac{8^{10}.3^{30}}{7^{30}.4^{15}}\).

Ta có:

\(VT:\frac{4^{15}}{7^{30}}=\frac{\left(2^2\right)^{15}}{7^{30}}=\frac{2^{30}}{7^{30}}\)

\(VP:\frac{8^{10}\cdot3^{30}}{7^{30}.4^{15}}=\frac{\left(2^3\right)^{10}.3^{30}}{7^{30}.\left(2^2\right)^{15}}=\frac{2^{30}.3^{30}}{7^{30}.2^{30}}=\frac{3^{30}}{7^{30}}\)

Ta thấy :\(\frac{2^{30}}{7^{30}}vs\frac{3^{30}}{7^{30}}\)có:

\(\orbr{\begin{cases}2^{30}< 3^{30}\\7^{30}=7^{30}\end{cases}\Rightarrow\frac{2^{30}}{7^{30}}< \frac{3^{30}}{7^{30}}\Leftrightarrow\frac{4^{15}}{7^{30}}< \frac{8^{10}.3^{30}}{7^{30}.4^{15}}}\)

Chúc bn hok tốt

< minh khong viet cach giai

\(\frac{4^{15}}{7^{30}}=\frac{2^{30}}{7^{30}}\)  và \(\frac{8^{10}.3^{30}}{7^{30}.4^{15}}=\frac{2^{30}.3^{30}}{7^{30}.2^{30}}=\frac{2^{30}}{7^{30}}\)Vậy hai vế bằng nhau 

Ta có:

\(\frac{4^{15}}{7^{30}}=\frac{\left(2^2\right)^{15}}{7^{30}}=\frac{2^{30}}{7^{30}}=\left(\frac{2}{7}\right)^{30}\)

\(\frac{8^{10}x3^{30}}{7^{30}x4^{15}}=\frac{\left(2^3\right)^{10}x3^{30}}{7^{30}x\left(2^2\right)^{15}}=\frac{2^{30}x3^{30}}{7^{30}x2^{30}}=\frac{3^{30}}{7^{30}}=\left(\frac{3}{7}\right)^{30}\)

Nhận thấy, 2 số đều có cùng số mũ mà \(\frac{3}{7}>\frac{2}{7}\)

=> \(\frac{8^{10}x3^{30}}{7^{30}x4^{15}}>\frac{4^{15}}{7^{30}}\)

Ta có: \(\frac{4^{15}}{7^{30}}\)=\(\frac{\left(2^2\right)^{15}}{7^{30}}\)=\(\frac{2^{30}}{7^{30}}\)=\(\frac{\left(2^3\right)^{10}}{7^{30}}\)=\(\frac{8^{10}}{7^{30}}\)

\(\frac{8^{10}.3^{10}}{7^{30}.4^{15}}\)=\(\frac{\left(2^3\right)^{10}.3^{10}}{7^{30}.\left(2^2\right)^{15}}\)=\(\frac{2^{30}.3^{10}}{7^{30}.2^{30}}\)=\(\frac{3^{10}}{7^{30}}\)

Vì 8¹⁰>3¹⁰ \(\Rightarrow\)\(\frac{8^{10}}{7^{30}}\)>\(\frac{3^{10}}{7^{30}}\)

Hay \(\frac{4^{15}}{7^{30}}\)>\(\frac{8^{10}.3^{10}}{7^{30}.4^{15}}\)

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