Question

simplify $4sqrt3{7} + 9sqrt3{56}$
$\circ$ $4sqrt3{7} + 9sqrt3{56}$
$\circ$ $13sqrt3{7}$
$\circ$ $22sqrt3{7}$
$\circ$ $36sqrt3{7}$

Explanation:

Step1: Factor 56 to simplify the cube root

We know that \( 56 = 8\times7 \), and \( \sqrt[3]{8}=2 \) because \( 2^3 = 8 \). So we can rewrite \( \sqrt[3]{56} \) as \( \sqrt[3]{8\times7} \). By the property of cube roots \( \sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b} \) (where \( a = 8 \) and \( b = 7 \)), we get \( \sqrt[3]{56}=\sqrt[3]{8}\times\sqrt[3]{7}=2\sqrt[3]{7} \).

Step2: Substitute the simplified cube root back into the expression

The original expression is \( 4\sqrt[3]{7}+9\sqrt[3]{56} \). Substituting \( \sqrt[3]{56} = 2\sqrt[3]{7} \) into the expression, we have \( 4\sqrt[3]{7}+9\times(2\sqrt[3]{7}) \).

Step3: Simplify the expression

First, calculate \( 9\times2 = 18 \), so the expression becomes \( 4\sqrt[3]{7}+18\sqrt[3]{7} \). Now, we can combine like terms. Since both terms have \( \sqrt[3]{7} \), we add the coefficients: \( (4 + 18)\sqrt[3]{7}=22\sqrt[3]{7} \).

Answer:

\( 22\sqrt[3]{7} \) (which corresponds to the option "22\(\boldsymbol{\sqrt[3]{7}}\)")