write the following in simplified radical form. $sqrt3{128}$

Explanation:

Step1: Prime - factorize 128

$128 = 2\times2\times2\times2\times2\times2\times2=2^{7}$

Step2: Rewrite the cube - root

$\sqrt[3]{128}=\sqrt[3]{2^{7}}$

Step3: Use the property of exponents $\sqrt[n]{a^{m}}=a^{\frac{m}{n}}$

$\sqrt[3]{2^{7}} = 2^{\frac{7}{3}}$

Step4: Rewrite the exponent as a mixed number

$\frac{7}{3}=2\frac{1}{3}$, so $2^{\frac{7}{3}}=2^{2+\frac{1}{3}}$

Step5: Use the property $a^{m + n}=a^{m}\times a^{n}$

$2^{2+\frac{1}{3}}=2^{2}\times2^{\frac{1}{3}}$

Step6: Simplify

$2^{2}\times2^{\frac{1}{3}} = 4\sqrt[3]{2}$

Answer:

$4\sqrt[3]{2}$