Combine the fractions using a common denominator.

Question

ScanSolve AI Tutor · Accepted Answer

Let's knock this out.
To simplify the expression, we will find a common denominator and combine the fractions.

Combine the fractions using a common denominator.
The common denominator is the product of the two denominators, $(\sqrt{17} - 2\sqrt{5})(\sqrt{17} + 2\sqrt{5})$ .
$\frac{1}{\sqrt{17} - 2\sqrt{5}} - \frac{1}{\sqrt{17} + 2\sqrt{5}} = \frac{(\sqrt{17} + 2\sqrt{5}) - (\sqrt{17} - 2\sqrt{5})}{(\sqrt{17} - 2\sqrt{5})(\sqrt{17} + 2\sqrt{5})}$
Simplify the numerator.
$(\sqrt{17} + 2\sqrt{5}) - (\sqrt{17} - 2\sqrt{5}) = \sqrt{17} + 2\sqrt{5} - \sqrt{17} + 2\sqrt{5}$ $= ( \sqrt{17} - \sqrt{17} ) + ( 2\sqrt{5} + 2\sqrt{5} )$ $= 0 + 4\sqrt{5}$ $= 4\sqrt{5}$
Simplify the denominator using the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$ .
Here, $a = \sqrt{17}$ and $b = 2\sqrt{5}$ .
$(\sqrt{17} - 2\sqrt{5})(\sqrt{17} + 2\sqrt{5}) = (\sqrt{17})^2 - (2\sqrt{5})^2$ $= 17 - (2^2 \cdot (\sqrt{5})^2)$ $= 17 - (4 \cdot 5)$ $= 17 - 20$ $= -3$
Substitute the simplified numerator and denominator back into the expression.
$\frac{4\sqrt{5}}{-3}$
Write the final simplified expression.
$-\frac{4\sqrt{5}}{3}$

Answer