\(\sqrt{1+\dfrac{1}{a^2}+\dfrac{1}{\left(a+1\right)^2}}=\sqrt{1+\left(\dfrac{1}{a^2}-\dfrac{2}{a^2\left(a+1\right)^2}+\dfrac{1}{\left(a+1\right)^2}\right)+\dfrac{2}{a^2\left(a+1\right)^2}}\)

\(=\sqrt{1+\dfrac{2}{a^2\left(a+1\right)^2}+\left(\dfrac{1}{a}-\dfrac{1}{a+1}\right)^2}\)

\(=\sqrt{1+\dfrac{2}{\left[a\left(a+1\right)\right]^2}+\dfrac{1}{\left[a\left(a+1\right)\right]^2}}=\sqrt{\left(1+\dfrac{1}{a\left(a+1\right)}\right)^2}\)

\(=\left|1+\dfrac{1}{a^2+1}\right|\)

Do \(a^2\ge0\Leftrightarrow a^2+1>0\Rightarrow\dfrac{1}{a^2+1}>0\Rightarrow\left|1+\dfrac{1}{a^2+1}\right|=1+\dfrac{1}{a^2+1}\)

\(\dfrac{1}{2!}+\dfrac{2}{3!}+\dfrac{3}{4!}+.....+\dfrac{2008}{2009!}=\dfrac{2-1}{2!}+\dfrac{3-1}{3!}+\dfrac{4-1}{4!}+.....+\dfrac{2009-1}{2009!}\)

\(=\dfrac{2}{2!}-\dfrac{1}{2!}+\dfrac{3}{3!}-\dfrac{1}{3!}+.....+\dfrac{2009}{2009!}-\dfrac{1}{2009!}\)

\(=\dfrac{1}{1!}-\dfrac{1}{2!}+\dfrac{1}{2!}-\dfrac{1}{3!}+.......+\dfrac{1}{2008!}-\dfrac{1}{2009!}\)

\(=1-\dfrac{1}{2009!}\)

a) ĐKXĐ: \(\left\{{}\begin{matrix}x\ne1\\x>0\end{matrix}\right.\)

b)

\(D=\left(\dfrac{2x+1}{\sqrt{x^3}-1}-\dfrac{\sqrt{x}}{x+\sqrt{x}+1}\right)\left(\dfrac{1+\sqrt{x^3}}{1+\sqrt{x}}-\sqrt{x}\right)\)

\(=\left(\dfrac{2x+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\right)\left(\dfrac{\left(1+\sqrt{x}\right)\left(1-\sqrt{x}+x\right)}{1+\sqrt{x}}-\sqrt{x}\right)\)

\(=\dfrac{2x+1-x+\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\left(1-\sqrt{x}+x-\sqrt{x}\right)\)

\(=\dfrac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\left(\sqrt{x}-1\right)^2\)

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

c)

Giả sử \(D>\dfrac{-2}{\sqrt{x}}\)

\(\Rightarrow\sqrt{x}-1>-\dfrac{2}{\sqrt{x}}\Leftrightarrow\sqrt{x}-1+\dfrac{2}{\sqrt{x}}>0\)

\(\Leftrightarrow\dfrac{\left(\sqrt{x}-1\right)\sqrt{x}+2}{\sqrt{x}}>0\Leftrightarrow x-\sqrt{x}+2>0\Leftrightarrow\left(x-\sqrt{x}+\dfrac{1}{4}\right)+\dfrac{7}{4}>0\Leftrightarrow\left(\sqrt{x}-\dfrac{1}{2}\right)^2+\dfrac{7}{4}>0\)(luôn đúng)

Find and correct the mistable.Choose the word that needs correcting:

1.What time does Mai has English class?

=> has -> have

2.What is Lan's favorite subject?

=> Subject => subjects

3.Do you often go at school library when you have free time ?

=> at -> to

4.She cleans the floor every morning.She clean it now.

=> clean -> is cleaning

5.We don't stay up late in night.

=> in -> at

6.When time do you usually get up every day?

=> When => what

7.Does your mother and Mrs.Lanh work here?Yes,they do

=> Does -> do

8.How many child does she have ?Two sons and two daughters.

=> Child -> children

I. Rewrite as directed.

1. Thanh is a fluent English speaker.

- Thanh speaks...................English fluently............................ .

2. What's your weight?

- How...........................much do you weigh?................................ .

3. Hoa has a big breakfast. Nam has a big breakfast. (so)

- ..................Hoa has a big breakfast, so does Nam...............................

4. He is a safe car driver.

- He drives............car safely............... .

5.My teacher drives a car to school every day.

- My teacher goes...........to school every day by car.................... .

6. Tim is tha tallest in his family.

- No one in Tim's family................is taller than Tim.................. .

7. I spend two hours a day doing my homework.

- It takes......................me 2 hours a day to do my homework...................... .

8. He is a slow and careful driver.

- He drives .................slowly and carefully......................

9. They mustn't stay up late. You mustn't stay up late. ( neither)

- ..................They mustn't stay up late, neither do you.........................

10. Nam is a hard student

- Nam............studies hard................. .

II. Read the passage and then answer the question.

Yesterday was Hoa's thirteenth birthday. She had a small party with her friends. They ate a lot of fool, fruits and cakes. In the evening, Hoa had a stomachache. Her aunt was worried. She called a doctor. The dpctor arrived and gave Hoa some medicine. She took the medicine and went to bed. Today, Hoa feels better.

1. When was Hoa's thirteenth birthday?

-......................It was yesterday..........................

2. Why did Hoa have a stomachache?

-......................Because she ate a lot of food, fruits and cakes ...................

3. Was her aunt worried?

-..........................Yes, she wasn't...................................

4. Does Hoa feel better today?

-....................Yes, she does...................