Value of Expression Calculation

The problem asks for the value of the expression $x^3 + y^3 + 3x^3y^3$ given the definitions of $x$ and $y$. We are given:

• $x = \sqrt[3]{31 + 14\sqrt{3}}$
• $y = \sqrt[3]{31 - 14\sqrt{3}}$

First, we find the values of $x^3$ and $y^3$ directly by cubing the given expressions:

• $x^3 = (\sqrt[3]{31 + 14\sqrt{3}})^3 = 31 + 14\sqrt{3}$
• $y^3 = (\sqrt[3]{31 - 14\sqrt{3}})^3 = 31 - 14\sqrt{3}$

Next, we calculate the sum $x^3 + y^3$:

$x^3 + y^3$ = $(31 + 14\sqrt{3}) + (31 - 14\sqrt{3})$

$x^3 + y^3$ = $31 + 31 = 62$

Now, we calculate the term $x^3y^3$. This can be written as $(xy)^3$. Let's find the value of $xy$ first:

$xy$ = $\sqrt[3]{31 + 14\sqrt{3}} \times \sqrt[3]{31 - 14\sqrt{3}}$

Using the property $\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab}$:

$xy$ = $\sqrt[3]{(31 + 14\sqrt{3})(31 - 14\sqrt{3})}$

Apply the difference of squares formula $(a+b)(a-b) = a^2 - b^2$ to the terms inside the cube root:

$xy$ = $\sqrt[3]{31^2 - (14\sqrt{3})^2}$

Calculate the squares:

$31^2$ = 961

$(14\sqrt{3})^2$ = $14^2 \times (\sqrt{3})^2 = 196 \times 3 = 588$

Substitute these values back:

$xy$ = $\sqrt[3]{961 - 588}$

$xy$ = $\sqrt[3]{373}$

Now, find $x^3y^3$:

$x^3y^3$ = $(xy)^3 = (\sqrt[3]{373})^3 = 373$

Finally, substitute the values of $x^3 + y^3$ and $x^3y^3$ into the expression $x^3 + y^3 + 3x^3y^3$:

Expression = $(x^3 + y^3) + 3(x^3y^3)$

Expression = $62 + 3(373)$

Expression = $62 + 1119$

Expression = $1181$