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
- Given that $7^{0.07} = x$, $7^{0.07} = y$ and $x^z = y^3$, then the value of z is close to:
- Find the value of $57^{-8} \div 57^{17} \times 57^{-18}$.
- If $\sqrt{45} + \sqrt{125} = 17.89$, then what will be the value of $\sqrt{180} + \sqrt{80}$?
(Give your answer to one decimal place.) If $\sqrt{2916} = 54$ then what is the value of the following?
$\sqrt{29.16} + \sqrt{0.2916} + \sqrt{0.002916} + \sqrt{0.00002916}$
- Find the value of $95^{-17} \div 95^{9} \times 95^{-15}$.
- $(\sqrt{5} + \sqrt{11})(\sqrt{5} - \sqrt{11})$ is equal to:
- What is the value of $\sqrt{75} + \sqrt{147}$?
- Find $\sqrt{\frac{1521}{1849}}$
If $\frac{2\sqrt{2}+\sqrt{7}}{2\sqrt{2}-\sqrt{7}} = x + y\sqrt{14}$, find the value of y.
- If $x = 7 + 4\sqrt{3}$, then find the value of $\sqrt{x} + \frac{1}{\sqrt{x}}$
Important Questions from Surds and Indices
Find the cube root of 78402752
Find the value of :
[(3 × 3 × 3 × 3 × 3 × 3) 6 ÷ (3 × 3 × 3 × 3) 7 × 3 4]
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 = ?
