Question

What is the simplest form of radical 1134

Answer 1

To simplify the square root of 1134, we need to factor 1134 into its prime factors.

1. First, we can divide 1134 by 2 (the smallest prime number):
\[
1134 \div 2 = 567
\]

2. Next, we factor 567:
\[
567 \div 3 = 189
\]

3. Continuing with 189:
\[
189 \div 3 = 63
\]

4. Now factor 63:
\[
63 \div 3 = 21
\]

5. Finally, factor 21:
\[
21 \div 3 = 7
\]

Putting this all together, the prime factorization of 1134 is:
\[
1134 = 2 \times 3^3 \times 7
\]

Now, we can simplify the square root:
\[
\sqrt{1134} = \sqrt{2 \times 3^3 \times 7}
\]

Since \(3^3\) can be rewritten as \(3^2 \times 3\), we can take \(3^2\) out of the square root:
\[
\sqrt{1134} = \sqrt{2 \times 3^2 \times 3 \times 7} = 3 \sqrt{2 \times 3 \times 7}
\]

Now calculate \(2 \times 3 \times 7\):
\[
2 \times 3 = 6
\]
\[
6 \times 7 = 42
\]

So, we have:
\[
\sqrt{1134} = 3 \sqrt{42}
\]

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