Question

station 1 simplifying radicals
simplify the following radical:
$sqrt3{250}$
$3sqrt3{2}$
$6sqrt3{2}$
$4sqrt3{2}$
$10sqrt3{2}$
$10$
$5sqrt3{2}$

Explanation:

Step1: Factor 250

We need to factor 250 into a product of a perfect cube and another number. We know that \(250 = 125\times2\), and \(125 = 5^3\). So, \(\sqrt[3]{250}=\sqrt[3]{5^3\times2}\).

Step2: Use the property of radicals

The property of radicals states that \(\sqrt[n]{ab}=\sqrt[n]{a}\times\sqrt[n]{b}\) (for \(a\geq0,b\geq0\) and \(n\) a positive integer). So, \(\sqrt[3]{5^3\times2}=\sqrt[3]{5^3}\times\sqrt[3]{2}\).

Step3: Simplify \(\sqrt[3]{5^3}\)

Since \(\sqrt[3]{a^3}=a\) for any real number \(a\), \(\sqrt[3]{5^3} = 5\). Therefore, \(\sqrt[3]{5^3}\times\sqrt[3]{2}=5\sqrt[3]{2}\).

Answer:

\(5\sqrt[3]{2}\) (corresponding to the option with \(5\sqrt[3]{2}\))