1. evaporates

2. condenses

3. precipitation

4. vapour 

5. surface

6. solid

7. liquid

1. water returns to the sea

2. evaporation

3. condensation         

4. precipitation            

Ta có: 

\(\left\{\begin{matrix}5^{27}=\left(5^3\right)^9=125^9\\2^{63}=\left(2^7\right)^9=128^9\end{matrix}\right\}\Rightarrow5^{27}< 2^{63}\left(1\right)\)

\(\left\{\begin{matrix}2^{63}=\left(2^9\right)^7=512^7\\5^{28}=\left(5^4\right)^7=625^7\end{matrix}\right\}\Rightarrow2^{63}< 5^{28}\left(2\right)\)

Từ (1) và (2) \(\Rightarrow5^{27}< 2^{63}< 5^{28}\) (đpcm)

a. ĐK: x \(\ge\) 0

b. ĐK = x \(\ge\) 0

\(Q=\dfrac{x+3}{\sqrt{x}+1}=\sqrt{x}-1+\dfrac{4}{\sqrt{x}+1}=\sqrt{x}+1+\dfrac{4}{\sqrt{x}+1}-2\)

Áp dụng bất đẳng thức Côsi với 2 biểu thức dương \(\sqrt{x}+1\) và \(\dfrac{4}{\sqrt{x+1}}\), ta được:

\(\sqrt{x}+1+\dfrac{4}{\sqrt{x}+1}\ge2\sqrt{\left(\sqrt{x}+1\right)\cdot\dfrac{4}{\sqrt{x}+1}}\)

\(\sqrt{x}+1+\dfrac{4}{\sqrt{x}+1}-2\ge2\sqrt{4}-2=4-2=2\)

\(P\ge2\)

Dấu "=" xảy ra khi \(\sqrt{x}+1=\dfrac{4}{\sqrt{x}+1}\)

\(\Leftrightarrow\left(\sqrt{x}+1\right)^2=4\)

\(\Leftrightarrow\left(\sqrt{x}+1\right)^2=2^2\)

\(\Leftrightarrow\sqrt{x}+1=2\)

\(\Leftrightarrow\sqrt{x}=1\)

\(\Leftrightarrow x=1\left(TM\right)\)

Vậy P_min =  2 khi x = 1

\((x+y)^2+(x-y)^2-2(x+y)(x-y)\)

\(= x^2 + 2xy + y^2 + x^2 - 2xy + y^2 - 2(x^2 - y^2)\)

\(= (x^2 + x^2) + (2xy - 2xy) + (y^2 + y^2) - 2x^2 + 2y^2\)

\(= 2x^2 + 2y^2 - 2x^2 + 2y^2\)

\(= (2x^2 - 2x^2) + (2y^2 + 2y^2)\)

\(= 4y^2\)

Ta có: \(P=\dfrac{\sqrt{x}-5}{\sqrt{x}+3}=1-\dfrac{8}{\sqrt{x}+3}\)

Ta có: \(\sqrt{x}\ge0\Leftrightarrow\sqrt{x}+3\ge3\)

\(\Rightarrow\dfrac{1}{\sqrt{x}+3}\le\dfrac{1}{3}\)

\(\Leftrightarrow-\dfrac{8}{\sqrt{x}+3}\ge-\dfrac{8}{3}\)

\(\Leftrightarrow1-\dfrac{8}{\sqrt{x}+3}\ge1-\dfrac{8}{3}=-\dfrac{5}{3}\)

\(\Leftrightarrow P\ge-\dfrac{5}{3}\)

Dấu "=" xảy ra khi x = 0

Vậy \(P_{min}=-\dfrac{5}{3}\) khi x = 0

\(\left(x+3\right)^2-\left(x-4\right)\cdot\left(x+4\right)=1\)

\(\Leftrightarrow x^2+6x+9-\left(x^2-16\right)=1\)

\(\Leftrightarrow x^2+6x+9-x^2+16=1\)

\(\Leftrightarrow x^2-x^2+6x=1-9-16\)

\(\Leftrightarrow6x=-24\)

\(\Leftrightarrow x=-4\)