Question

Radical 1,134 to simplest form

Answer 1

To simplify \(\sqrt{1134}\), we can start by factoring 1134 into its prime factors.

First, we can find its prime factorization:
\[
1134 \div 2 = 567
\]
So, \(1134 = 2 \times 567\).

Next, we can factor \(567\):
\[
567 \div 3 = 189
\]
So, \(567 = 3 \times 189\).

Now, we factor \(189\):
\[
189 \div 3 = 63
\]
So, \(189 = 3 \times 63\).

Next, we factor \(63\):
\[
63 \div 3 = 21
\]
So, \(63 = 3 \times 21\).

Finally, 21 can be factored as:
\[
21 \div 3 = 7
\]
So, \(21 = 3 \times 7\).

Now we can put all of this together:
\[
1134 = 2 \times 3^3 \times 7
\]

Next, we apply the square root:
\[
\sqrt{1134} = \sqrt{2 \times 3^3 \times 7}
\]
We can separate the perfect square (which is \(3^2\)):
\[
\sqrt{1134} = \sqrt{2 \times 3^2 \times 3 \times 7} = 3 \sqrt{2 \times 3 \times 7} = 3 \sqrt{42}
\]

Thus, the simplest form of \(\sqrt{1134}\) is:
\[
\sqrt{1134} = 3\sqrt{42}
\]