Simplify: √(250) – √(372) – √(-32) a. 8√2 – 6√3 – 4√2 b. 8√2 – 6√3 + 4√2 c. 8√2 – 6√3 – 4√5 d. 8√2 – 6√3 + 4√5
The Correct Answer and Explanation is:
To simplify the expression ( \sqrt{250} – \sqrt{372} – \sqrt{-32} ), we start by simplifying each square root individually.
- Simplifying ( \sqrt{250} ):
- We can factor ( 250 ) into ( 25 \times 10 ).
- This gives us:
[
\sqrt{250} = \sqrt{25 \times 10} = \sqrt{25} \cdot \sqrt{10} = 5\sqrt{10}.
]
- Simplifying ( \sqrt{372} ):
- We can factor ( 372 ) into ( 4 \times 93 ).
- This results in:
[
\sqrt{372} = \sqrt{4 \times 93} = \sqrt{4} \cdot \sqrt{93} = 2\sqrt{93}.
]
- Simplifying ( \sqrt{-32} ):
- The square root of a negative number introduces an imaginary unit ( i ). We factor ( -32 ) as ( -1 \times 32 ).
- Therefore, we have:
[
\sqrt{-32} = \sqrt{-1} \cdot \sqrt{32} = i \cdot \sqrt{32}.
] - Further simplifying ( \sqrt{32} ) gives:
[
\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}.
] - Thus,
[
\sqrt{-32} = 4i\sqrt{2}.
]
Putting it all together, we can rewrite the expression as:
[
\sqrt{250} – \sqrt{372} – \sqrt{-32} = 5\sqrt{10} – 2\sqrt{93} – 4i\sqrt{2}.
]
Now we need to check which of the answer choices aligns with this expression:
- None of the choices ( a, b, c, ) or ( d ) contain ( \sqrt{10} ) or ( \sqrt{93} ), suggesting they might have simplified incorrectly.
- However, ( \sqrt{-32} ) contributes an imaginary component ( 4i\sqrt{2} ), making options (a, b, c, d) that lack an imaginary term not applicable.
Conclusion
Given that we derived ( 5\sqrt{10} – 2\sqrt{93} – 4i\sqrt{2} ), and considering the presence of an imaginary number, the expression cannot be directly equated to any of the provided options. Thus, the provided options do not correspond correctly to the simplified form of the expression.