Question

write the expression so the radicand has no perfect squares other than 1?
$5\sqrt{75}$

$5\sqrt{75} = \square$
(type an exact answer, using radicals as needed. simplify your answer.)

Explanation:

Step1: Factor the radicand

We know that \(75 = 25\times3\), and \(25\) is a perfect square (\(25 = 5^2\)). So we can rewrite \(\sqrt{75}\) as \(\sqrt{25\times3}\).

Step2: Use the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a = 25\) and \(b = 3\)), we get \(\sqrt{25\times3}=\sqrt{25}\times\sqrt{3}\). Since \(\sqrt{25} = 5\), this simplifies to \(5\sqrt{3}\).

Step3: Multiply by the coefficient outside the radical

The original expression is \(5\sqrt{75}\), and we just found that \(\sqrt{75}=5\sqrt{3}\). So we multiply \(5\) (the coefficient outside) by \(5\sqrt{3}\): \(5\times5\sqrt{3}=25\sqrt{3}\).

Answer:

\(25\sqrt{3}\)