What is the simplified form of the radical expression √363 - 3√27?
To simplify the radical expression √363 - 3√27, you need to simplify both radicands (the numbers inside the square roots).
First, let's simplify √363. To do this, you need to find the prime factorization of 363. Start by dividing it by the smallest prime number, which is 3:
363 ÷ 3 = 121
Now, divide the result by 3 again:
121 ÷ 11 = 11
Since 11 is a prime number, we can stop here. The prime factorization of 363 is 3 × 3 × 11.
Now, let's simplify √27. Apply the same process, starting by dividing it by 3:
27 ÷ 3 = 9
Then, divide the result by 3 again:
9 ÷ 3 = 3
Again, 3 is a prime number, so we stop here. The prime factorization of 27 is 3 × 3 × 3.
Now, substitute the simplified versions of the radicands back into the expression:
√363 - 3√27 = √(3 × 3 × 11) - 3√(3 × 3 × 3)
Take the square root of each simplified radicand:
= √(3^2 × 11) - 3√(3^2 × 3)
Simplify further:
= 3√11 - 3 × 3√3
= 3√11 - 9√3
Therefore, the simplified form of √363 - 3√27 is 3√11 - 9√3.
To simplify the expression √363 - 3√27, we need to find the prime factors of the numbers under the radicals.
First, let's find the prime factors of 363:
363 can be divided by 3, resulting in 121.
121 can be divided by 11, resulting in 11.
So the prime factorization of 363 is 3 * 11 * 11.
Next, let's find the prime factors of 27:
27 can be divided by 3, resulting in 9.
9 can be divided by 3, resulting in 3.
So the prime factorization of 27 is 3 * 3 * 3.
Now, let's simplify the expression:
√363 - 3√27 = √(3 * 11 * 11) - 3√(3 * 3 * 3)
= √(3^2 * 11) - 3√(3^3)
= (3√11) - (3 * 3√3)
= 3√11 - 9√3
Therefore, the simplified form of the expression √363 - 3√27 is 3√11 - 9√3.