To express \(\sqrt{90}\) in its simplest radical form, follow these steps:
1. Prime Factorization of 90:
- Begin by breaking down 90 into its prime factors.
- 90 is an even number, so divide by 2: \(90 \div 2 = 45\).
- Next, factor 45. It's divisible by 3: \(45 \div 3 = 15\).
- Again, factor 15 by 3: \(15 \div 3 = 5\).
- Finally, 5 is a prime number.
So, the prime factorization of 90 is:
[tex]\[
90 = 2 \times 3 \times 3 \times 5 = 2 \times 3^2 \times 5
\][/tex]
2. Applying the Square Root:
- Use the property of square roots that states \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).
- So, apply this property to the prime factorization of 90:
[tex]\[
\sqrt{90} = \sqrt{2 \times 3^2 \times 5}
\][/tex]
3. Simplifying:
- Separate the perfect square from the other factors:
[tex]\[
\sqrt{2 \times 3^2 \times 5} = \sqrt{3^2 \times 2 \times 5}
\][/tex]
- \(\sqrt{3^2}\) simplifies to 3 because \(\sqrt{3^2} = 3\).
This leaves us with:
[tex]\[
3 \times \sqrt{2 \times 5} = 3 \times \sqrt{10}
\][/tex]
Therefore, the simplest radical form of \(\sqrt{90}\) is:
[tex]\[
3\sqrt{10}
\][/tex]
