Simplify the following.
​$$\frac{(289× 289× 289+ 111× 111× 111)}{(289× 289 –289× 111+ 111× 111)}$$​​

Correct option is A

​Given:

​$$\frac{(289 \times 289 \times 289 + 111 \times 111 \times 111)}{(289 \times 289 - 289 \times 111 + 111 \times 111)}$$​​

Concept Used:

$$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$ 

Solution:​

$$\frac{(289 \times 289 \times 289 + 111 \times 111 \times 111)}{(289 \times 289 - 289 \times 111 + 111 \times 111)}$$ ​

Let:

a =289 and b= 111

Thus, the expression becomes:

​$$\frac{(a \times a \times a + b \times b \times b)}{(a \times a - a \times b + b \times b)}$$  

The numerator is:

​$$a×a×a+b×b×b=a ^3 +b ^3$$ 

​Using the identity for the sum of cubes:

​$$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$ 

Thus, the numerator becomes: $$(a + b)(a^2 - ab + b^2)$$​​

The denominator is:

​$$a \times a - a \times b + b \times b = a^2 - ab + b^2$$​​

Now, substitute the simplified numerator and denominator into the original expression:

​$$\frac{(a + b)(a^2 - ab + b^2)}{a^2 - ab + b^2}$$ = (a + b)​

Now, that a=289 and b=111. Thus, the simplified expression is:

289 + 111 = 400

The simplified value of the expression is: 400.