To simplify the radical [tex]\(\sqrt{288}\)[/tex], we need to express it in its simplest radical form. Let's break this problem down step-by-step.
1. Factorize the number inside the square root:
The number 288 can be written as a product of prime factors:
[tex]\[
288 = 2^5 \times 3^2
\][/tex]
This factorization tells us that the prime factors of 288 are 2 raised to the power of 5 and 3 raised to the power of 2.
2. Group the factors to determine the simplest radical form:
We need to separate the factors into pairs because each pair of a factor can be taken outside the square root.
- For [tex]\(2^5\)[/tex]:
- There are two pairs of 2's ([tex]\(2^2 \times 2^2\)[/tex]) and one 2 left inside ([tex]\(2^1\)[/tex]).
- Each pair of 2's can be brought outside the square root as a single 2.
- Therefore, [tex]\(2^5\)[/tex] splits into [tex]\(2^2 \times 2^2 \times 2\)[/tex], which means we bring [tex]\(2 \times 2 = 4\)[/tex] outside the square root.
- For [tex]\(3^2\)[/tex]:
- There is one pair of 3's ([tex]\(3^2\)[/tex]).
- This pair of 3's can be brought outside the square root as a single 3.
- Therefore, [tex]\(3^2\)[/tex] brings out a single 3.
3. Combine the factors outside and inside the square root:
- The combined factors brought outside the square root are:
[tex]\[
4 \times 3 = 12
\][/tex]
- The remaining factor inside the square root is:
[tex]\[
\sqrt{2}
\][/tex]
4. Write the simplified radical form:
Combining the factors outside and inside the square root, we get:
[tex]\[
\sqrt{288} = 12 \sqrt{2}
\][/tex]
Therefore, the correct simplest radical form of [tex]\(\sqrt{288}\)[/tex] is:
[tex]\[
\boxed{12 \sqrt{2}}
\][/tex]
