Sure, let's simplify [tex]\(\sqrt{90}\)[/tex] step by step!
### Step 1: Prime Factorization
First, we need to factorize 90 into its prime factors.
- 90 is an even number, so it can be divided by 2:
[tex]\[
90 \div 2 = 45
\][/tex]
- 45 is divisible by 3 (since the sum of its digits, 4 + 5 = 9, is divisible by 3):
[tex]\[
45 \div 3 = 15
\][/tex]
- 15 is also divisible by 3:
[tex]\[
15 \div 3 = 5
\][/tex]
- Finally, 5 is a prime number and cannot be divided further.
So, the prime factorization of 90 is:
[tex]\[
90 = 2 \times 3^2 \times 5
\][/tex]
### Step 2: Simplify the Square Root
Using the property of square roots that [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we can split the square root as follows:
[tex]\[
\sqrt{90} = \sqrt{2 \times 3^2 \times 5}
\][/tex]
### Step 3: Extract Perfect Squares
Identify and extract the perfect squares from under the square root. Here, [tex]\(3^2\)[/tex] is a perfect square:
[tex]\[
\sqrt{2 \times 3^2 \times 5} = \sqrt{2} \times \sqrt{3^2} \times \sqrt{5}
\][/tex]
Since [tex]\(\sqrt{3^2} = 3\)[/tex]:
[tex]\[
\sqrt{2 \times 3^2 \times 5} = 3 \times \sqrt{2 \times 5} = 3 \times \sqrt{10}
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{90}\)[/tex] is:
[tex]\[
\sqrt{90} = 3 \times \sqrt{10}
\][/tex]
### Numerical Approximation
To provide a full understanding, let's also consider the numerical approximation of [tex]\(3 \times \sqrt{10}\)[/tex]:
[tex]\[
3 \times \sqrt{10} \approx 3 \times 3.162277660168379
\][/tex]
[tex]\[
3 \times 3.162277660168379 \approx 9.486832980505138
\][/tex]
Hence, [tex]\(\sqrt{90} \approx 9.486832980505138\)[/tex] as a numerical value.
### Summary
The fully simplified form of [tex]\(\sqrt{90}\)[/tex] is:
[tex]\[
\sqrt{90} = 3 \times \sqrt{10}
\][/tex]
And approximately:
[tex]\[
\sqrt{90} \approx 9.486832980505138
\][/tex]
