Simplify the radical expression. Show all your steps.
√363−3√27
To simplify the given radical expression √363 - 3√27, we need to simplify the radicals separately.
Let's start by simplifying √363.
To simplify a radical, we look for the largest perfect square that is a factor of the number inside the radical. We can see that 363 is divisible by 121 (11 × 11), which is a perfect square. Therefore, we can write 363 as 121 × 3:
√363 = √(121 × 3)
Next, we can use the property of radicals that states √(a × b) = √a × √b. Applying this property, we can break down the expression further:
√(121 × 3) = √121 × √3
The square root of 121 is 11:
√121 = 11
So, √363 becomes:
√363 = 11√3
Now, let's simplify the second part of the expression, which is -3√27.
We follow a similar process as before. We look for the largest perfect square that is a factor of 27. In this case, 27 is divisible by 9 (3 × 3), which is a perfect square. We can rewrite 27 as 9 × 3:
-3√27 = -3√(9 × 3)
Using the property of radicals again, we break it down further:
-3√(9 × 3) = -3√9 × √3
The square root of 9 is 3:
√9 = 3
So, -3√27 simplifies to:
-3√27 = -3(3√3) = -9√3
Now we combine the simplified radicals:
√363 - 3√27 = 11√3 - 9√3
Since both terms have the same radical (√3), we can combine them by subtracting their coefficients:
11√3 - 9√3 = (11 - 9)√3 = 2√3
Therefore, the simplified radical expression is 2√3.