b) $5^m=625\iff 5^m=5^4\iff m=4$.

c) By  ${16}^x{.5}^{2x}={16}^x{.25}^x={\left(16.25\right)}^x={400}^x$, so the done equation equivalant to   

                                                          ${400}^x={400}^1\iff x=1$.

d) $2^{2n+2}+2^{2n}=160\iff 2^{2n}\left(2^2+1\right)=160\iff 2^{2n}=2^5\iff 2n=5\iff n=2,5.$

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⇒x2 + 4x + 4 =24

=> x2 + 4x = 20

=> x(x + 4) = 20

=> x ,x + 4 thuộc Ư(20) = {1;2;4;5;10;20}

 When x = 1 thì x + 4 = 20 => x = 16 

Not yes value satisfies

Number of female students is :

  40 x 2525= 16 ( student )

Number of male students is :

   40 - 16 = 24 ( student )

      Answer : 24 student

  ( 1000 - 1 ) : 2 + 1 = 500,5 

The sum is :

  ( 1000 + 1 ) x 500,5 : 2 = 250500,25

We have : 

2a+b+c+da=a+2b+c+db=a+b+2c+dc=a+b+c+2dd2a+b+c+da=a+2b+c+db=a+b+2c+dc=a+b+c+2dd

⇒2a+b+c+da−1=a+2b+c+db−1=a+b+2c+dc−1=a+b+c+2dd−1⇒2a+b+c+da−1=a+2b+c+db−1=a+b+2c+dc−1=a+b+c+2dd−1

⇒a+b+c+da=a+b+c+db=a+b+c+dc=a+b+c+dd⇒a+b+c+da=a+b+c+db=a+b+c+dc=a+b+c+dd

+> If a + b + c + d ≠0≠0

=> a = b = c = d (Because the same of namerator)

=> M = 1 + 1 + 1 + 1 = 4

+> If a + b + c + d = 0

=> a + b = -(c + d)

      b + c = -(a + d)

      c + d = -(a + b)

      a + d = -(b + c)

Change this into M , we have :

M=a+bc+d+b+cd+a+c+da+b+a+db+cM=a+bc+d+b+cd+a+c+da+b+a+db+c

M=−(c+d)c+d+−(a+d)a+d+−(a+b)a+b+−(b+c)b+c=(−1)+(−1)+(−1)+(−1)=−4M=−(c+d)c+d+−(a+d)a+d+−(a+b)a+b+−(b+c)b+c=(−1)+(−1)+(−1)+(−1)=−4

So M = 4 and M = -4

 (x + 1 ) . 2000 = 4000

   x + 1 = 4000 : 2000

   x + 1 = 2 

   x = 2 - 1 

  x = 1 

Number of male students is :   50 - 20 = 30 ( students )

                                               Answer : 30 students

o be able to buy 9 bottles of fruit juice at the cheapest price, they should appoint 4 people to buy 8 bottles for: (7 + 1) x 4 = 32 (dollars)

The total amount they will pay to buy 9 bottles is: 32 + 7 = 39 (dollars)

We have : 

2a+b+c+da=a+2b+c+db=a+b+2c+dc=a+b+c+2dd2a+b+c+da=a+2b+c+db=a+b+2c+dc=a+b+c+2dd

⇒2a+b+c+da−1=a+2b+c+db−1=a+b+2c+dc−1=a+b+c+2dd−1⇒2a+b+c+da−1=a+2b+c+db−1=a+b+2c+dc−1=a+b+c+2dd−1

⇒a+b+c+da=a+b+c+db=a+b+c+dc=a+b+c+dd⇒a+b+c+da=a+b+c+db=a+b+c+dc=a+b+c+dd

+> If a + b + c + d ≠0≠0

=> a = b = c = d (Because the same of namerator)

=> M = 1 + 1 + 1 + 1 = 4

+> If a + b + c + d = 0

=> a + b = -(c + d)

      b + c = -(a + d)

      c + d = -(a + b)

      a + d = -(b + c)

Change this into M , we have :

M=a+bc+d+b+cd+a+c+da+b+a+db+cM=a+bc+d+b+cd+a+c+da+b+a+db+c

M=−(c+d)c+d+−(a+d)a+d+−(a+b)a+b+−(b+c)b+c=(−1)+(−1)+(−1)+(−1)=−4M=−(c+d)c+d+−(a+d)a+d+−(a+b)a+b+−(b+c)b+c=(−1)+(−1)+(−1)+(−1)=−4

So M = 4 and M = -

That worker makes 3 tools out of time is:

                          12 hours - 7 hours 30 minute = 4 hours 30 minute = 4,5 hours

The worker makes 1 product time out is :  

                          4,5 : 3 = 1,5 (hours)

                                 Answer : 1,5 hours

Put A = 1.2 + 2.3 + 3.4 + ....... + 199.200

 => 3A = 1.2.3 - 1.2.3 + 2.3.4 - 2.3.4 + ....... + 199.200.201

=> 3A = 199.200.201

=> A = 199.200.201/3

=> A = 2666600

Put A = 1.2 + 2.3 + 3.4 + ....... + 199.200

 => 3A = 1.2.3 - 1.2.3 + 2.3.4 - 2.3.4 + ....... + 199.200.201

=> 3A = 199.200.201

=> A = 199.200.201/3

=> A = 2666600

$\frac{2a+b+c+d}{a}=\frac{a+2b+c+d}{b}=\frac{a+b+2c+d}{c}=\frac{a+b+c+2d}{d}$

$\Rightarrow \frac{2a+b+c+d}{a}-1=\frac{a+2b+c+d}{b}-1=\frac{a+b+2c+d}{c}-1=\frac{a+b+c+2d}{d}-1$

$\Rightarrow \frac{a+b+c+d}{a}=\frac{a+b+c+d}{b}=\frac{a+b+c+d}{c}=\frac{a+b+c+d}{d}$

a) With 2x + 3 = x + 2

=> 2x - x = 2 - 3

=> x = -1

With 2x + 3 = -(x + 2)

=> 2x + 3 = -x - 2

=> 2x + x = -2 - 3

=> 3x = -5

=> x = −53

So x = -1 and x = −53

b) We have : 

A = |x - 2006| + |2007 - x| ≥ |x - 2006 + 2007 - x| = |1| = 1

⇔{|x−2006|≤0|2007−x|≤0⇒{x=2006x=2007

So when x = 2006 ; 2007 then value of A smallest 

We have : (x + 2)(x + 2) = 24(x + 2)(x + 2) = 24

⇒x2 + 4x + 4 =24

=> x2 + 4x = 20

=> x(x + 4) = 20

=> x ,x + 4 thuộc Ư(20) = {1;2;4;5;10;20}

 When x = 1 thì x + 4 = 20 => x = 16 

Not yes value satisfies

We have ( x + 1 ) . ( 1 + 1999 ) = 4000

 (x + 1 ) . 2000 = 4000

   x + 1 = 4000 : 2000

   x + 1 = 2 

   x = 2 - 1 

  x = 1 

We have : (x + 2)(x + 2) = 24(x + 2)(x + 2) = 24

⇒x2 + 4x + 4 =24

=> x2 + 4x = 20

=> x(x + 4) = 20

=> x ,x + 4 thuộc Ư(20) = {1;2;4;5;10;20}

 When x = 1 thì x + 4 = 20 => x = 16 

Not yes value satisfies

Number of female students is : 

   40 x 2525= 16 ( student )

 Number of  male students is :

    40 - 16 = 24 ( student )

         Answer : 24 student