1/16 ^ 10 : 1/4 ^ 15
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1.
a) \(3^4\times3^5\times3^6=3^{4+5+6}=3^{15}\)
b) \(5^2\times5^4\times5^5\times25=5^2\times5^4\times5^5\times5^2=5^{2+4+5+2}=5^{13}\)
c) \(10^8\div10^3=10^{8-3}=10^5\)
d) \(a^7\div a^2=a^{7-2}=a^5\)
2.
\(987=900+80+7\\ =9\times100+8\times10+7\\ =9\times10^2+8\times10^1+7\times10^0\)
\(2021=2000+20+1\\ =2\times1000+2\times10+1\times1\\ =2\times10^3+2\times10^1+1\times10^0\)
\(abcde=a\times10000+b\times1000+c\times100+d\times10+e\times1\\ =a\times10^4+b\times10^3+c\times10^2+d\times10^1+e\times10^0\)

a/ (26.16) :42
=(26.22) :(22)2
=28 : 24=24=16
b/ (510:625) : 25
= (510:54) :52
=56 : 52=53=625
c/ (81.243):3
= (34.35) :3
= 39 :3=38 =6561

`(1 1/4)^10 . (2/5)^20`
`=(5/4)^10 . (2/5)^20`
`=(5^10 .2^20)/(4^10 .5^20)`
`=(5^10 .4^10)/(4^10 .5^20)`
`=1/(5^10)`
`=(1/5)^10`

2^(10 : 64 x 16) = 2^[(10 x 16) : 64] = 2^(160 : 64) = 2^25
2^(10 x 8) = 2^80 3^(5 : 27) = 3^(5/27) 5^(2 x 125) = 5^250 6^(6 : 36) = 6^(1/6)

\(13^4\cdot16^2=13\cdot13\cdot13\cdot13\cdot16\cdot16\)
\(=13\cdot13\cdot13\cdot13\cdot4\cdot4\cdot4\cdot4\)
\(=\left(13\cdot4\right)\left(13\cdot4\right)\left(13\cdot4\right)\left(13\cdot4\right)\)
\(=52\cdot52\cdot52\cdot52\)
\(=52^4\)

2 ^ 10 . 15 + 2 ^ 10 . 85
= 2 ^ 10 . ( 15 + 55 )
= 2 ^ 10 . 100
= 1024 . 100
= 102400
= 320 ^ 2
\(\frac{1}{16^{10}}\) : \(\frac{1}{4^{15}}\)
= \(\frac{1}{\left(4^2\right)^{10}}\) : \(\frac{1}{4^{15}}\)
= \(\frac{1}{4^{20}}\) x \(\frac{4^{15}}{1}\)
= \(\frac{1}{4^5}\)
= 4\(^{-5}\)
1/16^10 ; 1/4^15
=1/(4^2)^10 . 4^15
=1/4^20 .4^15
=1/4^5