Question

Using suitable identity, evaluate the following:

`(92 xx 92 xx 92 + 8 xx 8 xx 8)/(92 xx 92 - 92 xx 8 + 8 xx 8)`

Answer 1

Given expression: `(92 xx 92 xx 92 + 8 xx 8 xx 8)/(92 xx 92 - 92 xx 8 + 8 xx 8)`

Step-wise calculation:

1. Rewrite the expression using powers:

`(92^3 + 8^3)/(92^2 - 92 xx 8 + 8^2)`

2. Recognize the denominator as a² – ab + b² with a = 92 and b = 8.

3. Use the cubic identity:

a³ + b³ = (a + b)(a² – ab + b²)

4. Substitute in the expression:

`(a^3 + b^3)/(a^2 - ab + b^2) = ((a + b)(a^2 - ab + b^2))/(a^2 - ab + b^2)`

`(a^3 + b^3)/(a^2 - ab + b^2) = a + b`

5. Therefore `(92^3 + 8^3)/(92^2 - 92 xx 8 + 8^2)` = 92 + 8 = 100.