Question

(1 / 7 - 4√3)^3 + (1 / 7 + 4√3)^3

Answer 1

Given:

The expression is:

$\left(\dfrac{1}{7}-4\sqrt{3}\right)^3+\left(\dfrac{1}{7}+4\sqrt{3}\right)^3$

To find:

The simplified form of the given expression.

Solution:

Formulae used:

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

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

Adding this formulae, we get

$(a-b)^3+(a+b)^3=2a^3+6ab^2$            ...(i)

We have,

$\left(\dfrac{1}{7}-4\sqrt{3}\right)^3+\left(\dfrac{1}{7}+4\sqrt{3}\right)^3$

Using formula (i), the given expression can be written as:

$=2\left(\dfrac{1}{7}\right)^3+6\left(\dfrac{1}{7}\right)\left(4\sqrt{3}\right)^2$

$=2\times \dfrac{1}{343}+6\left(\dfrac{1}{7}\right)48$

$=\dfrac{2}{343}+\dfrac{288}{7}$

$=\dfrac{2+14112}{343}$

$=\dfrac{14114}{343}$

Therefore, the simplified form of the given expression is $\dfrac{14114}{343}$.