`1)2^7*8`

`=2^7*2^3`

`=2^10`

`2)2^10*8`

`=2^10*2^3`

`=2^13`

`3)3^15*9`

`=3^15*3^2`

`=3^17`

`4)6^4*36`

`=6^4*6^2`

`=6^6`

`5)2^20:8`

`=2^20:2^3`

`=2^17`

`6)4^20:16`

`=4^20:4^2`

`=4^18`

`7)3^5:27`

`=3^5:3^3`

`=3^2`

`8)5^8:25`

`=5^8:5^2`

`=5^6` 

\(\sqrt{\dfrac{x^2}{4}+\sqrt{x^2-4}}=8-x^2\left(\left[{}\begin{matrix}2\le x\le2\sqrt{2}\\-2\sqrt{2}\le x\le-2\end{matrix}\right.\right)\\ \Leftrightarrow\sqrt{\dfrac{x^2+4\sqrt{x-4}}{4}}=8-x^2\\ \Leftrightarrow\sqrt{\dfrac{\left(x^2-4\right)+4\sqrt{x-4}+4}{4}}=8-x^2\\ \Leftrightarrow\sqrt{\left(\dfrac{\sqrt{x^2-4}+2}{2}\right)^2}=8-x^2\\ \Leftrightarrow\dfrac{\sqrt{x^2-4}+2}{2}=8-x^2\left(\dfrac{\sqrt{x^2-4}+2}{2}>0\right)\\ \Leftrightarrow\sqrt{x^2-4}+2=16-2x^2\\ \Leftrightarrow\sqrt{x^2-4}=14-2x^2\\ \Leftrightarrow x^2-4=\left(14-2x^2\right)^2\\ \Leftrightarrow x^2-4=4x^4-56x^2+196\\ \Leftrightarrow4x^4-57x^2+200=0\\ \Leftrightarrow\left(2x-5\right)\left(2x+5\right)\left(x^2-8\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=\dfrac{5}{2}\left(tm\right)\\x=-\dfrac{5}{2}\left(tm\right)\\x=\pm2\sqrt{2}\left(tm\right)\end{matrix}\right.\)

Góc `xPQ` và góc `uQn` 

Góc `mPy` và góc `mQv`

Góc `QPy` và góc `nQv` 

Ta có:

`B=-x^2+x-1`

`=(-x^2+x-1/4)-3/4`

`=-(x^2-x+1/4)-3/4`

`=-[x^2-2*x*1/2+(1/2)^2]-3/4`

`=-(x-1/2)^2-3/4`

Mà: `(x-1/2)^2>=0` với mọi x

`=>-(x-1/2)^2<=0` với mọi x

`=>-(x-1/2)^2-3/4<0` với mọi x 

`=>ĐPCM` 

`1)4^x=8^4`

`=>(2^2)^x=(2^3)^4`

`=>2^(2x)=2^12`

`=>2x=12`

`=>x=6`

`2)4^x=32^24`

`=>(2^2)^x=(2^5)^24`

`=>2^(2x)=2^(120)`

`=>2x=120`

`=>x=60`

`3)8^x=16^12`

`=>(2^3)^x=(2^4)^12`

`=>2^(3x)=2^48`

`=>3x=48`

`=>x=48/3`

`=>x=16` 

`4)4^x=8^12`

`=>(2^2)^x=(2^3)^12`

`=>2^(2x)=2^36`

`=>2x=36`

`=>x=18` 

`4/(1 xx 10) + 4/(10 xx 19) + ... + 4/(18 xx 91)`

`=4(1/(1 xx 10)+1/(10 xx 19)+..+1/(18 xx 91))`

`=4/9*(9/(1 xx 10) + 9/(10 xx 19) + ...+9/(18 xx 91))`

`=4/9(1-1/10+1/10-1/19+...+1/18-1/91)`

`=4/9*(1-1/91)`

`=4/9*90/91`

`=40/91` 

Bài 6:

\(a+b+c=0\Leftrightarrow\left\{{}\begin{matrix}a+b=-c\\a+c=-b\\b+c=-a\end{matrix}\right.\)

\(\Rightarrow P=\dfrac{\left(-c\right)^2+\left(-a\right)^2+\left(-b\right)^2}{a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ca+a^2}\\ =\dfrac{a^2+b^2+c^2}{2\left(a^2+b^2+c^2-ab-bc-ca\right)}\)

Mà: `a+b+c=0`

`=>(a+b+c)^2=0`

`=>a^2+b^2+c^2+2ab+2bc+2ca=0`

`=>-ab-bc-ca=1/2(a^2+b^2+c^2)`

\(P=\dfrac{a^2+b^2+c^2}{2\left[a^2+b^2+c^2-\dfrac{1}{2}\left(a^2+b^2+c^2\right)\right]}\\=\dfrac{a^2+b^2+c^2}{2\cdot\dfrac{1}{2}\cdot\left(a^2+b^2+c^2\right)}=1\)

`S=(1-1/2)*(1-1/3)*...*(1-1/2024)`

`=(2/2-1/2)*(3/3-1/3)*...*(2024/2024-1/2024)`

`=1/2*2/3*...*2023/2024`

`=(2*3*...*2023)/((2*3*...*2023)*2024)`

`=1/2024` 

Bài 3: 

a) `a+b=6`

`<=>a=6-b`

Mà: `a^2+b^2=20`

`=>(6-b)^2+b^2=20`

`<=>36-12b+b^2+b^2=20`

`<=>2b^2-12b+36=20`

`<=>b^2-6b+18=10`

`<=>b^2-6b+8=0`

`<=>(b^2-2b)+(-4b+8)=0`

`<=>(b-2)(b-4)=0`

`<=>b=2` hoặc `b=4` 

Với `b=2=>a=6-2=4`

Với `b=4=>a=6-4=2` 

b) `a+b=100=>a=100-b`

Mà: `ab=2500`

`=>(100-b)b=2500`

`<=>100b-b^2=2500`

`<=>b^2-100b=2500`

`<=>(b-50)^2=0`

`<=>b=50`

`=>a=100-50=50` 

`(x-4)/5+(3x-2)/10-x=(2x-5)/3-(7x+2)/6`

`<=>(2(x-4))/10+(3x-2)/10-(10x)/10=(2(2x-5))/6-(7x+2)/6`

`<=>(2x-8+3x-2-10x)/10=(4x-10-7x-2)/6`

`<=>(-5x-10)/10=(-3x-12)/6`

`<=>(-5(x+2))/10=(-3(x+4))/6`

`<=>(x+2)/-2=(x+4)/-2`

`<=>x+2=x+4`

`<=>2=4`

Điều này vô lý 

=> Không có x thỏa mãn