Sure, let's break down the problem step-by-step.
1. Prime Factorization of the Radicand:
- The number under the square root is called the radicand. In this case, the radicand is 135.
- Prime factorization of 135:
[tex]\[\begin{align*}
135 &= 3 \times 45 \\
45 &= 3 \times 15 \\
15 &= 3 \times 5
\end{align*}\][/tex]
Thus, the prime factorization of 135 is
[tex]\[135 = 3^3 \times 5\][/tex]
2. Simplified Form of [tex]\( 3 \sqrt{135} \)[/tex]:
- We start with [tex]\( 3 \sqrt{135} \)[/tex].
- Writing 135 using its prime factors, we get
[tex]\[3 \sqrt{135} = 3 \sqrt{3^3 \times 5}\][/tex]
- We can rewrite the radicand as
[tex]\[135 = 9 \times 15\][/tex]
because [tex]\( 9 = 3^2 \)[/tex] and it is a perfect square.
- Therefore,
[tex]\[3 \sqrt{135} = 3 \sqrt{9 \times 15}\][/tex]
- Applying the product property of square roots, which states that [tex]\( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)[/tex], we get
[tex]\[3 \sqrt{9 \times 15} = 3 \sqrt{9} \times \sqrt{15}\][/tex]
- We know [tex]\( \sqrt{9} = 3 \)[/tex], so this becomes:
[tex]\[3 \times 3 \times \sqrt{15} = 9 \sqrt{15}\][/tex]
Hence, the simplified form of [tex]\( 3 \sqrt{135} \)[/tex] is [tex]\( 9 \sqrt{15} \)[/tex].
3. Verifying with Product Property:
- Given expressions such as:
[tex]\[3 \sqrt{5(3)}=3 \sqrt{15}\][/tex]
are basic applications of rewriting numbers in terms of their factors.
- When we write it step-by-step:
[tex]\[3 \sqrt{135} = 3(3) \sqrt{5(3)} = 9 \sqrt{15}\][/tex]
- And this confirms the most simplified version of the expression.
Therefore, the final simplified form of [tex]\( 3 \sqrt{135} \)[/tex] is indeed [tex]\( 9 \sqrt{15} \)[/tex].
