\(81^7-27^9-9^{13}=\left(3^4\right)^7-\left(3^3\right)^9-\left(3^2\right)^{13}=3^{28}-3^{27}-3^{26}=3^{26}\left(3^2-3-1\right)=3^{24}\left(3^4-3^3-3^2\right)=3^{24}.45⋮45\)

Điều kiện của m là b, \(m\ne1\)

\(x-4+\sqrt{x^2}+8x+16\left(1\right)\)

TH1: \(0\le x< 4\)

\(\left(1\right)=x-4+x+8x+16=2x+8x+12\)

TH2: \(x< 0\)

\(\left(1\right)=x-4-x+8x+16=8x+12\)

Ta có: \(\overline{abcdeg}=1000\overline{abc}+\overline{deg}=2000\overline{deg}+\overline{deg}=2001\overline{deg}\)

Vì 2001 chia hết cho 23 và 29 \(\Rightarrow2001\overline{deg}\) chia hết cho 23 và 29

Vậy \(\overline{abcdeg}\) chia hết cho 23 và 29

\(\sqrt{23-8\sqrt{7}}=\sqrt{\left(\sqrt{16}-\sqrt{7}\right)^2}=\sqrt{16}-\sqrt{7}=4-\sqrt{7}\)

\(\sqrt{9-4\sqrt{5}}=\sqrt{\left(\sqrt{5}-\sqrt{4}\right)^2}=\sqrt{5}-\sqrt{4}=\sqrt{5}-2\)

1) \(\sqrt{15+4\sqrt{14}}=\sqrt{\left(\sqrt{7}+\sqrt{8}\right)^2}=\sqrt{7}+\sqrt{8}\)

2) \(\sqrt{15-4\sqrt{14}}+\sqrt{16-6\sqrt{7}}=\sqrt{\left(\sqrt{8}-\sqrt{7}\right)^2}+\sqrt{\left(\sqrt{9}-\sqrt{7}\right)^2}=\sqrt{8}-\sqrt{7}+\sqrt{9}-\sqrt{7}=3+2\sqrt{2}-2\sqrt{7}\)3) \(C=\sqrt{8+4\sqrt{3}}+\sqrt{15-6\sqrt{6}}=\sqrt{\left(\sqrt{6}+\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{9}-\sqrt{6}\right)^2}=\sqrt{6}+\sqrt{2}+\sqrt{9}-\sqrt{6}=3+\sqrt{2}\)4) \(D=\sqrt{4-2\sqrt{3}}-\sqrt{3}=\sqrt{\left(\sqrt{3}-1\right)^2}-\sqrt{3}=\sqrt{3}-1-\sqrt{3}=-1\)5) \(E=\sqrt{4-\sqrt{15}}+\sqrt{6-\sqrt{35}}-\sqrt{5-\sqrt{21}}+\sqrt{2}=\sqrt{\left(\sqrt{\dfrac{5}{2}}-\sqrt{\dfrac{3}{2}}\right)^2}+\sqrt{\left(\sqrt{\dfrac{7}{2}}-\sqrt{\dfrac{5}{2}}\right)^2}-\sqrt{\left(\sqrt{\dfrac{7}{2}}-\sqrt{\dfrac{3}{3}}\right)^2}+\sqrt{2}=\sqrt{\dfrac{5}{2}}-\sqrt{\dfrac{3}{2}}+\sqrt{\dfrac{7}{2}}-\sqrt{\dfrac{5}{2}}-\sqrt{\dfrac{7}{2}}+\sqrt{\dfrac{3}{2}}+\sqrt{2}=\sqrt{2}\)

\(C=ab\left(a+b\right)-bc\left(b+c\right)+ac\left(a-c\right)=ab\left(a+b\right)-bc\left(a+b-a+c\right)+ac\left(a-c\right)=ab\left(a+b\right)-bc\left(a+b\right)+bc\left(a-c\right)+ac\left(a-c\right)=b\left(a+b\right)\left(a-c\right)+c\left(a-c\right)\left(a+b\right)=\left(a+b\right)\left(c+c\right)\left(a-c\right)\)

\(B=a\left(b^3-c^3\right)+b\left(c^3-a^3\right)+c\left(a^3-b^3\right)=ab^3-ac^3+bc^3-a^3b+a^3c-b^3c=ab\left(b^2-a^2\right)-c^3\left(a-b\right)+c\left(a^3-b^3\right)=-ab\left(a-b\right)\left(a+b\right)-c^3\left(a-b\right)+c\left(a-b\right)\left(a^2+ab+b^2\right)=\left(a-b\right)\left(-a^2b-ab^2-c^3+a^2c+abc+b^2c\right)\)

\(A=x\left(y^2-z^2\right)+y\left(z^2-x^2\right)+z\left(x^2-y^2\right)=x\left(y^2-z^2\right)+y\left(-y^2+z^2-x^2+y^2\right)+z\left(x^2-y^2\right)=\left(y^2-z^2\right)\left(x-y\right)+\left(x^2-y^2\right)\left(z-y\right)=\left(y-z\right)\left(y+z\right)\left(x-y\right)-\left(x-y\right)\left(x+y\right)\left(y-z\right)=\left(x-y\right)\left(y-z\right)\left(y+z-x-y\right)=\left(x-y\right)\left(y-z\right)\left(z-x\right)\)