Question

what is the following product?
$sqrt{30} cdot sqrt{10}$
options:
$4sqrt{10}$
$3sqrt{10}$
$10sqrt{3}$
$2sqrt{10}$

Explanation:

Step1: Multiply the radicands

Using the property of square roots \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\), we have \(\sqrt{30}\cdot\sqrt{10}=\sqrt{30\times10}\)
\(\sqrt{30\times10}=\sqrt{300}\)

Step2: Simplify the square root

Factor 300 to find perfect square factors: \(300 = 100\times3\), so \(\sqrt{300}=\sqrt{100\times3}\)
Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (where \(a = 100\), \(b = 3\)), we get \(\sqrt{100\times3}=\sqrt{100}\cdot\sqrt{3}\)
Since \(\sqrt{100}=10\), this simplifies to \(10\sqrt{3}\)

Answer:

\(10\sqrt{3}\) (corresponding to the option with text "10√3")