According to the title, we have: a = 9

b = 8

So b + a : 0,5 = 8 + 9 : 0,5 = 8 + 18 = 26 :) 

x² = 6400 

\(\Rightarrow x=\pm\sqrt{6400}\)

\(\Rightarrow x=\pm80\)

* x = 80 => x isn't a square number

* x = -80 < 0 isn't  square number

So................

We have : x² = 6400

=> x² = \(\pm80^2\)

To try on:

- x = 80 => x it isn't a square number

- x = -80 < 0 it isn't square number

=> x it isn't square number

We have this equation{a=3ba−3=b+3

(with a is the length and b is the width)

⇒3b−3=b+3

⇒3b=b+6

⇒2b=6

⇒b=3

⇒a=9

So that the rug is 3m wide and 9m long.

We have this equation\(\left\{{}\begin{matrix}a=3b\\a-3=b+3\end{matrix}\right.\)(with a is the length and b is the width)

\(\Rightarrow3b-3=b+3\)

\(\Rightarrow3b=b+6\)

\(\Rightarrow2b=6\)

\(\Rightarrow b=3\)

\(\Rightarrow a=9\)

So that the rug is 3m wide and 9m long.

(a) Assume that the square number 16 = 4² is equal to\(\dfrac{n}{3}\),so n = 48

(b) Assume that the cubic number 27 = 3³ is equal to\(\dfrac{n}{5}\),so n = 135

(a) Assume that the square number 16 = 42 is equal ton3n3,so n = 48

(b) Assume that the cubic number 27 = 33 is equal ton5n5,so n = 135

 1 Selected by MathYou

(a) Assume that the square number 16 = 42 is equal ton3n3,so n = 48

(b) Assume that the cubic number 27 = 33 is equal ton5n5,so n = 135

 1 Selected by MathYou

The difference between 2 square numbers in the question is : 
(n + 19) - (n - 6) = 25

So those square numbers must be 0 and 25.Then that number is 6

The difference between 2 square numbers in the question is : 
(n + 19) - (n - 6) = 25

So those square numbers must be 0 and 25.Then that number is 6

Selected by MathYou

24a + 96b = 4(6a + 24b) 

4 is a square number,so 24a + 96b is a square number only when

6a + 24b is a square number

a,b are positive integers⇒a,b ≥ 1⇒ 6a + 24b ≥ 30

When a + b gets the least value,so does 6a + 24b.Hence,6a + 24b = 36

⇒24b < 36 ⇒24b = 24⇒{6a=12b=1⇒{a=2b=1

=> a + b = 3

So 3 is the least value of a + b

24a + 96b = 4(6a + 24b) 

4 is a square number,so 24a + 96b is a square number only when

6a + 24b is a square number

a,b are positive integers⇒a,b ≥ 1⇒ 6a + 24b ≥ 30

When a + b gets the least value,so does 6a + 24b.Hence,6a + 24b = 36

⇒24b < 36 ⇒24b = 24\(\Rightarrow\left\{{}\begin{matrix}6a=12\\b=1\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a=2\\b=1\end{matrix}\right.\)=> a + b = 3

So 3 is the least value of a + b

The difference between 2 above square numbers is :

(151 + n) - (100 + n) = 51

So those square numbers must be 49 and 100.Then n = -51

The difference between 2 above square numbers is :

(151 + n) - (100 + n) = 51

So those square numbers must be 49 and 100.Then n = -51

We have :

n(n + 1)(n + 2)(n + 3) + 1 = [n(n + 3)][(n + 1)(n + 2)] + 1

= (n2 + 3n)(n2 + 3n + 2) + 1 = (n2 + 3n)2 + 2.(n2 + 3n) + 12 

= (n2 + 3n + 1)2

Replace n = 2006 into the expression,we have :

2006 x 2007 x 2008 x 2009 + 1

= (20062 + 3 x 2006 + 1)2 = 40300552

We have :

n(n + 1)(n + 2)(n + 3) + 1 = [n(n + 3)][(n + 1)(n + 2)] + 1

= (n² + 3n)(n² + 3n + 2) + 1 = (n² + 3n)² + 2.(n² + 3n) + 1² 

= (n² + 3n + 1)²

Replace n = 2006 into the expression,we have :

2006 x 2007 x 2008 x 2009 + 1

= (2006² + 3 x 2006 + 1)² = 4030055²

The nth term is

¯¯¯¯¯¯¯¯¯¯¯(n)52=(10n+5)2=100n2+100n+25=100n(n+1)+25

=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯[n(n+1)]25

Example :

To find the 20th term,we calculate 20(20 + 1) = 420

So the answer is 42025

The n^th term is

\(\overline{\left(n\right)5}^2=\left(10n+5\right)^2=100n^2+100n+25=100n\left(n+1\right)+25\)

\(=\overline{\left[n\left(n+1\right)\right]25}\)

Example :

To find the 20^th term,we calculate 20(20 + 1) = 420

So the answer is 42025

The difference between 2 above square number is 184,so those number are 441 and 625.

Then x = 441

The difference between 2 above square number is 184,so those number are 441 and 625.

Then x = 441

You should check the question.This problem is easy

ab - ba = 0,so n2 = 0.Then n = 0

Do you think its 

501^2-500^2

a =3;a= -3

Ya :3 

Given a sum :  2 + 4 + 6 + ... + 2k

The number of terms is : 2k−22+1=k

 ( terms )

So, the result of this sum is : k.(2k+2)2=k.(k+1)

Example : 2 + 4 + 6 = 2 + 4 + 2.3 ; so k = 3; then 2 + 4 +6 = k.(k+1)=3.4 .

Given a sum :  2 + 4 + 6 + ... + 2k

The number of terms is : 2k−22+1=k2k−22+1=k ( terms )

So, the result of this sum is : k.(2k+2)2=k.(k+1)k.(2k+2)2=k.(k+1)

Example : 2 + 4 + 6 = 2 + 4 + 2.3 ; so k = 3; then 2 + 4 +6 = k.(k+1)=3.4 .

Given a sum :  2 + 4 + 6 + ... + 2k

The number of terms is : \(\dfrac{2k-2}{2}+1=k\) ( terms )

So, the result of this sum is : \(\dfrac{k.\left(2k+2\right)}{2}=k.\left(k+1\right)\)

Example : 2 + 4 + 6 = 2 + 4 + 2.3 ; so k = 3; then 2 + 4 +6 = k.(k+1)=3.4 .

We have : ab = 8(a + b) 

=> 10a + b = 8a + 8b

=> 10a - 8a = 8b - b

=> 2a = 7b

=> a = 7 ; b = 2

So ab is 72

We have : ab = 8(a + b) 

=> 10a + b = 8a + 8b

=> 10a - 8a = 8b - b

=> 2a = 7b

=> a = 7 ; b = 2

So ab is 72