Ta có:\(4^{30}=2^{30}.2^{30}=2^{30}.4^{15}>\left(2^3\right)^{10}.3^{15}=\left(8.3\right)^{10}.3^5>24^{10}.3\)

Do đó \(2^{30}+3^{30}+4^{30}>3.24^{10}\)

2³⁰+3³⁰+4³⁰=4¹⁵+27¹⁰+64¹⁰>4¹⁵+24¹⁰+2.24¹⁰>3.24¹⁰

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đặt A= 3.24¹⁰=3.(8.3)¹⁰=\(\frac{3}{2}\).2.2³⁰.(\(\frac{3}{2}\))¹⁰ . 2¹⁰ = \(\left(\frac{3}{2}\right)^{11}.2^{41}\)

đặt B=2³⁰+3³⁰ + 4³⁰ =2³⁰ +\(\left(\frac{3}{2}\right)^{30}.2^{30}+2^{60}\)

A-B= \(\frac{3}{2}^{11}.2^{41}-2^{30}-2^{30}.\left(\frac{3}{2}\right)^{30}-2^{60}\)

     =\(2^{30}\left(2^{11}.\left(\frac{3}{2}\right)^{11}-1-\left(\frac{3}{2}\right)^{30}-2^{30}\right)\)

     =\(2^{30}\left(3^{11}-1-\left(\frac{3}{2}\right)^{30}-2^{30}\right)A\)

nhớ ****

4^30=2^30*2^30

=2^30*4^15

3*24^10=3*3^10*8^10=3^11*2^30

mà 4^30>3^11

nên 2^30+3^30+4^30>3*24^10

Ta có: 4^30=2^30.2^30=2^30.4^15

3.24^10=3.(3.2^3)^10=2^30.3^11

Ta thấy: 3^11<3^15<4^15 => 4^15>3^11

Vì 4^15>3^11 nên 2^30.4^15>2^30.3^11

=>2^30+3^30+4^30>3.24^10

\(3\times24^{10}\)

\(=3\times\left(2^3\times3\right)^{10}\)

\(=3\times3^{10}\times\left(2^3\right)^{10}\)

\(=3^{11}\times2^{30}\)

\(=3^{11}\times\left(2^2\right)^{15}\)

\(=3^{11}\times4^{15}\)

Vì \(3^{11}\)<\(4^{15}\left(3;4;11;15\inℕ\right)\)

Nên \(3^{11}\times4^{15}\)< \(4^{15}\times4^{15}=4^{30}\)

Do đó : \(3\times24^{10}\)< \(4^{30}\)

Vậy \(2^{30}+3^{30}+4^{30}\)> \(3\times24^{10}\)