Simplifying the Radical Expression
The problem asks for the value of the expression $\frac{4+\sqrt{2}}{\sqrt{2}+1}$. To find the value, we need to simplify this expression, typically by rationalizing the denominator.
Rationalizing the Denominator
To rationalize the denominator $\sqrt{2}+1$, we multiply both the numerator and the denominator by its conjugate, which is $\sqrt{2}-1$.
The expression becomes:
$ \frac{4+\sqrt{2}}{\sqrt{2}+1} \times \frac{\sqrt{2}-1}{\sqrt{2}-1} $Calculating the Numerator
Multiply the numerators:
$ (4+\sqrt{2})(\sqrt{2}-1) $Using the distributive property (FOIL method):
$ 4(\sqrt{2}) + 4(-1) + \sqrt{2}(\sqrt{2}) + \sqrt{2}(-1) $ $ = 4\sqrt{2} - 4 + (\sqrt{2})^2 - \sqrt{2} $ $ = 4\sqrt{2} - 4 + 2 - \sqrt{2} $Combine like terms:
$ = (4\sqrt{2} - \sqrt{2}) + (-4 + 2) $ $ = 3\sqrt{2} - 2 $Calculating the Denominator
Multiply the denominators:
$ (\sqrt{2}+1)(\sqrt{2}-1) $This is in the form $(a+b)(a-b) = a^2 - b^2$. Here, $a = \sqrt{2}$ and $b = 1$.
$ = (\sqrt{2})^2 - (1)^2 $ $ = 2 - 1 $ $ = 1 $Final Simplification
Now, divide the simplified numerator by the simplified denominator:
$ \frac{3\sqrt{2} - 2}{1} $ $ = 3\sqrt{2} - 2 $Therefore, the value of the expression $\frac{4+\sqrt{2}}{\sqrt{2}+1}$ is $3\sqrt{2}-2$. This matches Option B.
Similar Questions
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Important Questions from Surds and Indices
The cube root of - 64 × - 1331 is:
If (27) m = (81) n, then m 2: mn = ?
if 49 n + 49 n + 49 n + 49 n + 49 n + 49 n + 49 n = 7 2221 , then n = ?
If a x= b y= c zand b 2= ac then y =
The value of (0.3) [{(200 - 146)/(3 × 3 × 3)} - 3] is:
