simplify 3√8 − √24 − √32 − 4√54
To start simplifying the expression, we can simplify the square roots individually.
First, let's simplify the square root of 8:
√8 = √(4 * 2) = 2√2
Next, let's simplify the square root of 24:
√24 = √(4 * 6) = 2√6
Now, let's simplify the square root of 32:
√32 = √(4 * 8) = 2√8 = 2(2√2) = 4√2
Lastly, let's simplify the square root of 54:
√54 = √(9 * 6) = 3√6
Now, we can substitute the simplified square roots back into the original expression:
3√8 − √24 − √32 − 4√54
= 3(2√2) - 2√6 - 4(3√6)
= 6√2 - 2√6 - 12√6
= (6√2 - 12√6) - 2√6
= 6√2 - (12√6 + 2√6)
= 6√2 - 14√6
Therefore, the simplified expression is 6√2 - 14√6.
To simplify the expression 3√8 − √24 − √32 − 4√54, we can start by simplifying the square roots.
Let's break down each of the square roots:
√8 = √(4 * 2) = 2 * √2
√24 = √(4 * 6) = 2 * √6
√32 = √(16 * 2) = 4 * √2
√54 = √(9 * 6) = 3 * √6
Now we can substitute these back into the original expression:
3√8 − √24 − √32 − 4√54
= 3(2 * √2) - (2 * √6) - (4 * √2) - 4(3 * √6)
= 6√2 - 2√6 - 4√2 - 12√6
= (6√2 - 4√2) + (-2√6 - 12√6)
= 2√2 - 14√6
Therefore, the simplified expression is 2√2 - 14√6.
To simplify the given expression, we need to simplify each individual square root term and then combine like terms.
Let's start by simplifying each square root term:
1. √8:
We can simplify this term by factoring out the perfect square from 8. The largest perfect square that divides 8 evenly is 4.
√8 = √(4 * 2) = √4 * √2 = 2√2
2. √24:
We can simplify this term in a similar way by factoring out the largest perfect square that divides 24 evenly. The largest perfect square that divides 24 is 4.
√24 = √(4 * 6) = √4 * √6 = 2√6
3. √32:
Again, we'll factor out the largest perfect square that divides 32. The largest perfect square that divides 32 is 16.
√32 = √(16 * 2) = √16 * √2 = 4√2
4. √54:
This term has a larger perfect square factor. The largest perfect square that divides 54 is 9.
√54 = √(9 * 6) = √9 * √6 = 3√6
Now, we can substitute the simplified terms back into the expression and combine like terms:
3√8 - √24 - √32 - 4√54
= 3(2√2) - 2√6 - 4(4√2) - 4(3√6)
= 6√2 - 2√6 - 16√2 - 12√6
Finally, we can combine the like terms:
6√2 - 16√2 = (6 - 16)√2 = -10√2
-2√6 - 12√6 = (-2 - 12)√6 = -14√6
Therefore, the simplified expression is:
-10√2 - 14√6.