Simplify the radicals and collect the like terms
√450 - √288 + √32
why stop there?
√18 + 4√2 = 3√2 + 4√2 = 7√2
Apologies for the oversight. You are absolutely correct.
Continuing from √18 + 4√2:
√18 + 4√2 = 3√2 + 4√2 = 7√2.
Therefore, the simplified expression is 7√2.
To simplify radicals and collect like terms, we need to find the factors of the numbers under the square roots that are perfect squares.
Let's start with √450. We can find the factors of 450 that are perfect squares:
√450 = √(9 * 50) = √(3^2 * 5^2 * 2) = 3 * 5√2 = 15√2
Next, let's simplify √288:
√288 = √(16 * 18) = √(4^2 * 18) = 4√18
Lastly, simplify √32:
√32 = √(16 * 2) = √(4^2 * 2) = 4√2
Now we can rewrite the expression with the simplified radicals:
15√2 - 4√18 + 4√2
To collect like terms, we combine the terms with the same radical:
(15√2 + 4√2) - 4√18
19√2 - 4√18
So the simplified expression with collected like terms is 19√2 - 4√18.
To simplify the radicals, we can find the largest perfect square that divides each number inside the square root.
For √450:
The largest perfect square that divides 450 is 25 since 25 * 18 = 450.
So √450 = √(25 * 18) = √25 * √18 = 5√18.
For √288:
The largest perfect square that divides 288 is 16 since 16 * 18 = 288.
So √288 = √(16 * 18) = √16 * √18 = 4√18.
For √32:
The largest perfect square that divides 32 is 16 since 16 * 2 = 32.
So √32 = √(16 * 2) = √16 * √2 = 4√2.
Now we can rewrite the expression:
√450 - √288 + √32 = 5√18 - 4√18 + 4√2
Finally, to collect like terms, we combine the terms with the same radical:
5√18 - 4√18 + 4√2 = (5 - 4)√18 + 4√2 = √18 + 4√2.
Therefore, the simplified expression is √18 + 4√2.