Question 23
Let $S$ be the set of ordered triples $(x, y, z)$ of real numbers for which $$
\log_{10}(x + y) = z
$$ and $$
\log_{10}(x^2 + y^2) = z + 1.
$$ There are real numbers $a$ and $b$ such that for all ordered triples $(x, y, z)$ in $S$ we have $x^3 + y^3 = a \cdot 10^{3z} + b \cdot 10^{2z}.$ What is the value of $a + b?$
\log_{10}(x + y) = z
$$ and $$
\log_{10}(x^2 + y^2) = z + 1.
$$ There are real numbers $a$ and $b$ such that for all ordered triples $(x, y, z)$ in $S$ we have $x^3 + y^3 = a \cdot 10^{3z} + b \cdot 10^{2z}.$ What is the value of $a + b?$