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## Find the value of the expression

### Solution

\begin{aligned} &a=8+3 \sqrt{21} \\\\ &b=8-3 \sqrt{21} \end{aligned}
\begin{aligned} x&=\sqrt[3]{a}+\sqrt[3]{b}\\\\ x^{3}&=(\sqrt[3]{a}+\sqrt[3]{b})^{3} \\\\ &=a+b+3(\sqrt[3]{a})^{2}(\sqrt[3]{b})+3(\sqrt[3]{b})^{2}(\sqrt[3]{a})\\\\ &=a+b+3 \sqrt[3]{a a b}+3 \sqrt[3]{b b a} \\\\ a b&=b a =(8+3 \sqrt{21})(8-3 \sqrt{21}) \\\\ &=8^{2}-3^{2}(21) \\\\ &=-125=(-5)^{3} \end{aligned}
\begin{aligned} x^{3} &=a+b+3(-5) \sqrt[3]{a}+3(-5) \sqrt[3]{b} \\\\ &=a+b-15(\sqrt[3]{a}+\sqrt[3]{b})\\\\ x&=\sqrt[3]{a}+\sqrt[3]{b} \\\\ x^{3}&=16-15x\\\\ x^{3}&+15x-16=0 \end{aligned}
Notice $$x=1$$ is a solution, Thus $$(x-1)$$ is a factor: $(x-1)\left(x^{2}+x+16\right)=0$
The quadratic roots involve imaginary parts and non-zero. We know that: $\begin{array}{ll} a=8+3 \sqrt{21} & \sqrt[3]{a} \in \mathbb{R} \\\\ b=8-3 \sqrt{21} & \sqrt[3]{b} \in \mathbb{R} \end{array}$ $\Rightarrow x=1$
Therefore $\large \sqrt[3]{8+3 \sqrt{21}}+\sqrt[3]{8-3 \sqrt{21}}=1$
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