Question

use the product rule to simplify the radical.
$4\sqrt{250}$
$4\sqrt{250}=\square$
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Factor 250 into perfect square and other

We know that \(250 = 25\times10\), where \(25\) is a perfect square (\(25 = 5^{2}\)). So we can rewrite \(\sqrt{250}\) as \(\sqrt{25\times10}\).

Step2: Apply the product rule for radicals

The product rule for radicals states that \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)). So \(\sqrt{25\times10}=\sqrt{25}\times\sqrt{10}\). Since \(\sqrt{25} = 5\), this becomes \(5\sqrt{10}\).

Step3: Multiply by the coefficient outside the radical

We have \(4\sqrt{250}=4\times\sqrt{25\times10}=4\times(\sqrt{25}\times\sqrt{10}) = 4\times5\times\sqrt{10}\).

Step4: Calculate the product of the coefficients

\(4\times5 = 20\), so \(4\sqrt{250}=20\sqrt{10}\).

Answer:

\(20\sqrt{10}\)