Question

simplify $8sqrt3{56} + 2sqrt3{7}$.

Explanation:

Step1: Factor 56 in the cube root

We know that \(56 = 8\times7\), and \(\sqrt[3]{8\times7}=\sqrt[3]{8}\times\sqrt[3]{7}\) because \(\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}\) (for real numbers \(a,b\)). Since \(\sqrt[3]{8} = 2\), then \(\sqrt[3]{56}=\sqrt[3]{8\times7}=2\sqrt[3]{7}\).

Step2: Substitute back into the original expression

The original expression is \(8\sqrt[3]{56}+2\sqrt[3]{7}\). Substituting \(\sqrt[3]{56}=2\sqrt[3]{7}\) into it, we get \(8\times(2\sqrt[3]{7})+2\sqrt[3]{7}\).

Step3: Simplify the first term

Calculate \(8\times(2\sqrt[3]{7})\), which is \(16\sqrt[3]{7}\). So now the expression is \(16\sqrt[3]{7}+2\sqrt[3]{7}\).

Step4: Combine like terms

Since both terms have \(\sqrt[3]{7}\), we can combine them. Using the distributive property \(ac + bc=(a + b)c\) (here \(c = \sqrt[3]{7}\), \(a = 16\), \(b = 2\)), we get \((16 + 2)\sqrt[3]{7}=18\sqrt[3]{7}\).

Answer:

\(18\sqrt[3]{7}\)