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\(a,125^5:25^3=\left(5^3\right)^5:\left(5^2\right)^3=5^{15-6}=5^9\)
\(b,27^6:9^3=\left(3^3\right)^6:\left(3^2\right)^3=3^{18-6}=3^{12}\)
\(c,4^{20}:2^{15}=\left(2^2\right)^{20}:2^{15}=2^{40-15}=2^{25}\)
\(d,24^n:2^{2n}=4^n.6^n:4^n=6^n\)
\(e,64^4.16^5:4^{20}=\left(2^6\right)^4.\left(2^4\right)^5:\left(2^2\right)^{20}=2^{24+20-40}=2^4=16\)
\(A=\left(-3\right)\times9\times\left(-8\right)\times5^6=3\times9\times8\times5^6=3^3\times2^3\times5^6\).
\(B=2+2+2^2+2^3+...+2^{11}\)
\(2B=2^2+2^2+2^3+2^4+...+2^{12}\)
\(2B-B=\left(4+2^2+2^3+2^4+...+2^{12}\right)-\left(2+2+2^2+2^3+...+2^{11}\right)\)
\(B=2^{12}\)
\(A=8^2\cdot32^4\\ =2^6\cdot2^{20}\\ =2^{26}\\ B=27^3\cdot9^4\cdot243\\ =3^9\cdot3^{12}\cdot3^5\\ =3^{26}\)
`A=8^2*32^4=(2^3)^2*(2^5)^4=2^6*2^20=2^26`
`B=27^3*9^4*243=(3^3)^3*(3^2)^4*3^5=3^9*3^8*3^5=3^22`
e. a 4 . a . a 5 . a 6 . a 8 = a 24