Question

question 1-28
what is the value of the expression $7\sqrt3{500} + 11\sqrt3{256}$ using only a single radical?
\bigcirc\\ $27\sqrt3{4}$
\bigcirc\\ $79\sqrt3{4}$
\bigcirc\\ $54\sqrt3{28}$
\bigcirc\\ $18\sqrt3{756}$

Explanation:

Step1: Simplify \(7\sqrt[3]{500}\)

Factor 500: \(500 = 125\times4 = 5^3\times4\). So, \(\sqrt[3]{500}=\sqrt[3]{5^3\times4}=5\sqrt[3]{4}\). Then \(7\sqrt[3]{500}=7\times5\sqrt[3]{4}=35\sqrt[3]{4}\).

Step2: Simplify \(11\sqrt[3]{256}\)

Factor 256: \(256 = 64\times4 = 4^3\times4\). So, \(\sqrt[3]{256}=\sqrt[3]{4^3\times4}=4\sqrt[3]{4}\). Then \(11\sqrt[3]{256}=11\times4\sqrt[3]{4}=44\sqrt[3]{4}\).

Step3: Add the two simplified terms

\(35\sqrt[3]{4}+44\sqrt[3]{4}=(35 + 44)\sqrt[3]{4}=79\sqrt[3]{4}\).

Answer:

\(79\sqrt[3]{4}\) (corresponding to the option "79\(\boldsymbol{\sqrt[3]{4}}\)")