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 $3\sqrt{135}$?
$\circ$ $\sqrt{15}$
$\circ$ $3\sqrt{5(3)} = 3\sqrt{15}$
$\circ$ $(3 + 3)\sqrt{5(3)} = 6\sqrt{15}$
$\circ$ $3(3)\sqrt{5(3)} = 9\sqrt{15}$

Explanation:

Step1: Prime factorization of 135

Prime factorize 135: \(135 = 3\times3\times3\times5\)

Step2: Apply product property of square roots

We know that \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)). We want to form perfect square factors. \(135 = 9\times15\) (since \(9 = 3\times3\) is a perfect square)
So \(3\sqrt{135}=3\sqrt{9\times15}\)

Step3: Simplify the square root

Since \(\sqrt{9} = 3\), then \(3\sqrt{9\times15}=3\times\sqrt{9}\times\sqrt{15}=3\times3\times\sqrt{15}=9\sqrt{15}\) (Wait, no, wait the original expression is \(3\sqrt{135}\), let's re - check. Wait, maybe I misread the original problem. Wait the original problem is \(3\sqrt{135}\)? Wait no, looking at the options, let's re - do:

Wait, prime factorization of 135: \(135=3\times3\times3\times5 = 9\times15=9\times3\times5\)? Wait no, \(135 = 3^3\times5\). Now, \(\sqrt{135}=\sqrt{9\times15}=\sqrt{9}\times\sqrt{15} = 3\sqrt{15}\). Then \(3\sqrt{135}=3\times3\sqrt{15}=9\sqrt{15}\)? Wait but let's check the options again. Wait the options are:

1. \(\sqrt{15}\)
2. \(3\sqrt{5(3)} = 3\sqrt{15}\)
3. \((3 + 3)\sqrt{5(3)}=6\sqrt{15}\)
4. \(3(3)\sqrt{5(3)} = 9\sqrt{15}\)

Wait, let's start over. The original expression is \(3\sqrt{135}\)

Prime factorize 135: \(135=3\times3\times3\times5=3^{3}\times5\)

\(\sqrt{135}=\sqrt{3^{2}\times3\times5}=\sqrt{3^{2}}\times\sqrt{3\times5}=3\sqrt{15}\)

Then \(3\sqrt{135}=3\times3\sqrt{15}=9\sqrt{15}\) which is the last option. But wait, maybe I made a mistake. Wait no, let's check the steps again.

Wait step 1: Prime factorization of 135: \(135 = 3\times3\times3\times5\)

Step 2: Write as product of perfect square and other factors: \(135=9\times15\) (since \(9 = 3\times3\))

Step 3: Simplify \(\sqrt{135}=\sqrt{9\times15}=\sqrt{9}\times\sqrt{15}=3\sqrt{15}\)

Then \(3\sqrt{135}=3\times3\sqrt{15}=9\sqrt{15}\)

So the correct option is the last one: \(3(3)\sqrt{5(3)} = 9\sqrt{15}\) (even though the way of writing \(5(3)\) is \(15\), it's a bit non - standard, but mathematically \(3(3)\sqrt{5\times3}=9\sqrt{15}\))

Wait, but let's check again. Wait maybe the original expression is \(\sqrt{135}\) multiplied by 3? Wait the problem says "What is the simplified form of \(3\sqrt{135}\)?"

So:

1. Prime factorization of 135: \(135 = 3\times3\times3\times5\)
2. Apply product property: \(\sqrt{135}=\sqrt{9\times15}\) (because \(9 = 3\times3\) is a perfect square)
3. Simplify: \(\sqrt{9\times15}=\sqrt{9}\times\sqrt{15}=3\sqrt{15}\)
4. Then \(3\sqrt{135}=3\times3\sqrt{15}=9\sqrt{15}\) which matches the last option \(3(3)\sqrt{5(3)} = 9\sqrt{15}\) (since \(5\times3 = 15\))

Answer:

\(3(3)\sqrt{5(3)} = 9\sqrt{15}\) (the last option: \(3(3)\sqrt{5(3)} = 9\sqrt{15}\))