Simplify: (sqrt(2) + i) / (sqrt(2) + sqrt(5))

Question

ScanSolve AI Tutor · Accepted Answer

To simplify the expression, we need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

Multiply the numerator and denominator by the conjugate of the denominator.
The denominator is $\sqrt{2}+\sqrt{5}$ , so its conjugate is $\sqrt{2}-\sqrt{5}$ .
$\frac{\sqrt{2}+1}{\sqrt{2}+\sqrt{5}} \times \frac{\sqrt{2}-\sqrt{5}}{\sqrt{2}-\sqrt{5}}$
Expand the numerator.
$(\sqrt{2}+1)(\sqrt{2}-\sqrt{5}) = \sqrt{2} \cdot \sqrt{2} - \sqrt{2} \cdot \sqrt{5} + 1 \cdot \sqrt{2} - 1 \cdot \sqrt{5}$ $= 2 - \sqrt{10} + \sqrt{2} - \sqrt{5}$
Expand the denominator using the difference of squares formula $(a+b)(a-b) = a^2 - b^2$ .
$(\sqrt{2}+\sqrt{5})(\sqrt{2}-\sqrt{5}) = (\sqrt{2})^2 - (\sqrt{5})^2$ $= 2 - 5$ $= -3$
Combine the expanded numerator and denominator.
$\frac{2 - \sqrt{10} + \sqrt{2} - \sqrt{5}}{-3}$
Rewrite the expression by moving the negative sign to the numerator or in front of the fraction.
$\frac{-(2 - \sqrt{10} + \sqrt{2} - \sqrt{5})}{3} = \frac{-2 + \sqrt{10} - \sqrt{2} + \sqrt{5}}{3}$
Rearranging the terms in the numerator:
$\frac{\sqrt{10} + \sqrt{5} - \sqrt{2} - 2}{3}$

Simplify: (sqrt(2) + i) / (sqrt(2) + sqrt(5))

Answer

Step-by-step solution

Multiply the numerator and denominator by the conjugate of the denominator.

Expand the numerator.

Expand the denominator using the difference of squares formula $(a+b)(a-b) = a^2 - b^2$ .

Combine the expanded numerator and denominator.

Rewrite the expression by moving the negative sign to the numerator or in front of the fraction.

Try one yourself

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Simplify: (sqrt(2) + i) / (sqrt(2) + sqrt(5))

Answer

Step-by-step solution

Multiply the numerator and denominator by the conjugate of the denominator.

Expand the numerator.

Expand the denominator using the difference of squares formula (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2(a+b)(a−b)=a2−b2.

Combine the expanded numerator and denominator.

Rewrite the expression by moving the negative sign to the numerator or in front of the fraction.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Expand the denominator using the difference of squares formula $(a+b)(a-b) = a^2 - b^2$ .