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19 Feb 2023, 01:13
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What is the value of \(|\sqrt{3} - |\sqrt{5}-\sqrt{7}|| - ||\sqrt{3} - \sqrt{5}|-\sqrt{7}|\) ?
A. \(2\sqrt{5}-2\sqrt{7}-2\sqrt{3}\)
B. \(2\sqrt{5}-2\sqrt{7}\)
C. \(2\sqrt{7}-2\sqrt{5}\)
D. \(2\sqrt{5}-2\sqrt{7}+2\sqrt{3}\)
E. \(2\sqrt{7}+2\sqrt{5}\)
A. \(2\sqrt{5}-2\sqrt{7}-2\sqrt{3}\)
B. \(2\sqrt{5}-2\sqrt{7}\)
C. \(2\sqrt{7}-2\sqrt{5}\)
D. \(2\sqrt{5}-2\sqrt{7}+2\sqrt{3}\)
E. \(2\sqrt{7}+2\sqrt{5}\)
19 Feb 2023, 01:13
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Official Solution:
What is the value of \(|\sqrt{3} - |\sqrt{5}-\sqrt{7}|| - ||\sqrt{3} - \sqrt{5}|-\sqrt{7}|\) ?
A. \(2\sqrt{5}-2\sqrt{7}-2\sqrt{3}\)
B. \(2\sqrt{5}-2\sqrt{7}\)
C. \(2\sqrt{7}-2\sqrt{5}\)
D. \(2\sqrt{5}-2\sqrt{7}+2\sqrt{3}\)
E. \(2\sqrt{7}+2\sqrt{5}\)
To answer this question we should recall the property of the absolute value:
\(|x| = x\), when \(x \geq 0\);
\(|x| = -x\), when \(x < 0\).
So, we should evaluate the expressions in the modulus to see whether they are positive or negative.
STEP 1:
Since \(\sqrt{5}-\sqrt{7} < 0\), then \(|\sqrt{5}-\sqrt{7}|=-(\sqrt{5}-\sqrt{7})=\sqrt{7}-\sqrt{5}\);
Since \(\sqrt{3} - \sqrt{5} < 0\), then \(|\sqrt{3} - \sqrt{5}|=-(\sqrt{3} - \sqrt{5})= \sqrt{5} -\sqrt{3}\).
Thus, \(|\sqrt{3} - |\sqrt{5}-\sqrt{7}|| - ||\sqrt{3} - \sqrt{5}|-\sqrt{7}|\) will become:
\(|\sqrt{3} -(\sqrt{7}-\sqrt{5})| - |(\sqrt{5} -\sqrt{3})-\sqrt{7}|=|\sqrt{3} +\sqrt{5}-\sqrt{7}| - |\sqrt{5} -\sqrt{3}-\sqrt{7}|\).
STEP 2:
\(\sqrt{3} +\sqrt{5}-\sqrt{7}\) must be positive because \(\sqrt{3} +\sqrt{5}=1.something + 2.something=3.something\), while \(\sqrt{7} < 3.something\). So, \(|\sqrt{3} +\sqrt{5}-\sqrt{7}| =\sqrt{3} +\sqrt{5}-\sqrt{7}\);
\(\sqrt{5} -\sqrt{3}-\sqrt{7}\) is obviously negative. So, \(\sqrt{5} -\sqrt{3}-\sqrt{7}=-(\sqrt{5} -\sqrt{3}-\sqrt{7})=\sqrt{7}+\sqrt{3}-\sqrt{5}\).
Thus, \(|\sqrt{3} +\sqrt{5}-\sqrt{7}| - |\sqrt{5} -\sqrt{3}-\sqrt{7}|\) will become:
\((\sqrt{3} +\sqrt{5}-\sqrt{7})-(\sqrt{7}+\sqrt{3}-\sqrt{5})=2\sqrt{5}-2\sqrt{7}\).
Answer: B
What is the value of \(|\sqrt{3} - |\sqrt{5}-\sqrt{7}|| - ||\sqrt{3} - \sqrt{5}|-\sqrt{7}|\) ?
A. \(2\sqrt{5}-2\sqrt{7}-2\sqrt{3}\)
B. \(2\sqrt{5}-2\sqrt{7}\)
C. \(2\sqrt{7}-2\sqrt{5}\)
D. \(2\sqrt{5}-2\sqrt{7}+2\sqrt{3}\)
E. \(2\sqrt{7}+2\sqrt{5}\)
To answer this question we should recall the property of the absolute value:
\(|x| = x\), when \(x \geq 0\);
\(|x| = -x\), when \(x < 0\).
So, we should evaluate the expressions in the modulus to see whether they are positive or negative.
STEP 1:
Since \(\sqrt{5}-\sqrt{7} < 0\), then \(|\sqrt{5}-\sqrt{7}|=-(\sqrt{5}-\sqrt{7})=\sqrt{7}-\sqrt{5}\);
Since \(\sqrt{3} - \sqrt{5} < 0\), then \(|\sqrt{3} - \sqrt{5}|=-(\sqrt{3} - \sqrt{5})= \sqrt{5} -\sqrt{3}\).
Thus, \(|\sqrt{3} - |\sqrt{5}-\sqrt{7}|| - ||\sqrt{3} - \sqrt{5}|-\sqrt{7}|\) will become:
\(|\sqrt{3} -(\sqrt{7}-\sqrt{5})| - |(\sqrt{5} -\sqrt{3})-\sqrt{7}|=|\sqrt{3} +\sqrt{5}-\sqrt{7}| - |\sqrt{5} -\sqrt{3}-\sqrt{7}|\).
STEP 2:
\(\sqrt{3} +\sqrt{5}-\sqrt{7}\) must be positive because \(\sqrt{3} +\sqrt{5}=1.something + 2.something=3.something\), while \(\sqrt{7} < 3.something\). So, \(|\sqrt{3} +\sqrt{5}-\sqrt{7}| =\sqrt{3} +\sqrt{5}-\sqrt{7}\);
\(\sqrt{5} -\sqrt{3}-\sqrt{7}\) is obviously negative. So, \(\sqrt{5} -\sqrt{3}-\sqrt{7}=-(\sqrt{5} -\sqrt{3}-\sqrt{7})=\sqrt{7}+\sqrt{3}-\sqrt{5}\).
Thus, \(|\sqrt{3} +\sqrt{5}-\sqrt{7}| - |\sqrt{5} -\sqrt{3}-\sqrt{7}|\) will become:
\((\sqrt{3} +\sqrt{5}-\sqrt{7})-(\sqrt{7}+\sqrt{3}-\sqrt{5})=2\sqrt{5}-2\sqrt{7}\).
Answer: B
21 Jan 2025, 03:26
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I like the solution - it’s helpful.
