Question

1. write the prime factorization of the radicand.
2. apply the product property of square roots. write the radicand as a product, forming as many perfect square roots as possible.
3. simplify.

what is the simplified form of ( 3sqrt{135} )?
options: ( sqrt{15} ); ( 3sqrt{5(3)} = 3sqrt{15} ); ( 3(3)sqrt{5(3)} = 9sqrt{15} ); ( (3 + 3)sqrt{5(3)} = 6sqrt{} )

Explanation:

Step1: Prime factorization of 135

Prime factorize 135: \(135 = 3\times3\times3\times5=9\times15\) (since \(9 = 3\times3\) is a perfect square)

Step2: Apply product property of square roots

Using \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we have \(3\sqrt{135}=3\sqrt{9\times15}\)

Step3: Simplify \(\sqrt{9}\)

Since \(\sqrt{9} = 3\), then \(3\sqrt{9\times15}=3\times\sqrt{9}\times\sqrt{15}=3\times3\times\sqrt{15}=9\sqrt{15}\)

Answer:

The simplified form of \(3\sqrt{135}\) is \(9\sqrt{15}\), so the correct option is the one with \(3(3)\sqrt{5(3)} = 9\sqrt{15}\) (the third box in the given options).