Question - Evaluating an Algebraic Expression Involving Square Roots
Solution:
Given the equation \(2\sqrt{2} + \sqrt{125} - \sqrt{45} + 4 = a + b\sqrt{c}\),
First, simplify the square roots that are not in simplest form:
\(\sqrt{125} = \sqrt{25 \cdot 5} = 5\sqrt{5}\),
\(\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}\).
Now, substitute these simplified forms back into the equation:\(2\sqrt{2} + 5\sqrt{5} - 3\sqrt{5} + 4 = a + b\sqrt{c}\).
Combine like terms:\(2\sqrt{2} + (5\sqrt{5} - 3\sqrt{5}) + 4 = a + b\sqrt{c}\),
\(2\sqrt{2} + 2\sqrt{5} + 4 = a + b\sqrt{c}\).
Now, match the terms on both sides of the equation:\(a = 4\),
\(b = 2\), \(c = 2\),
\(b = 2\), \(c = 5\).
Since the square root terms must match (they are like terms), we can see that there are two terms with a square root on the left side, \(\sqrt{2}\) and \(\sqrt{5}\). These two terms must match the single square root term on right side, \(b\sqrt{c}\). This is only consistent if \(c\) is a product of the radicands \(2\) and \(5\). So this suggests that \(b\) should be the sum of the coefficients of the square root terms on the left, while \(c\) should be the product of the radicands:\(b=2+2=4\),
\(c=2 \cdot 5 = 10\).
Hence, the equation can be written as:\(4 + 4\sqrt{10} = a + b\sqrt{c}\).
Now that \(a\), \(b\), and \(c\) have been found, evaluate \(2a - b\):\(2a - b = 2(4) - 4 = 8 - 4 = 4\).
The value of \(2a - b\) is \(4\).CamTutor
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