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(sqrt5-2)/(sqrt5+2)+(sqrt5+2)/(sqrt5-2)=a-bsqrt5

$\frac{\sqrt{5}-2}{\sqrt{5}+2}+\frac{\sqrt{5}+2}{\sqrt{5}-2}=a-b \sqrt{5}$

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Factor sums and differences of cubes

Full solution

Q.

\frac{\sqrt{5}-2}{\sqrt{5}+2}+\frac{\sqrt{5}+2}{\sqrt{5}-2}=a-b \sqrt{5}

Find Common Denominator: First, let's find a common denominator for the two fractions. $\newline$ The common denominator is $(\sqrt{5} + 2)(\sqrt{5} - 2)$ .
Multiply by Common Denominator: Now, let's multiply each fraction by the common denominator divided by its own denominator to get equivalent fractions with the common denominator. $\newline$ For the first fraction: $\left(\frac{\sqrt{5} - 2}{\sqrt{5} + 2}\right) * \left(\frac{\sqrt{5} - 2}{\sqrt{5} - 2}\right)$ $\newline$ For the second fraction: $\left(\frac{\sqrt{5} + 2}{\sqrt{5} - 2}\right) * \left(\frac{\sqrt{5} + 2}{\sqrt{5} + 2}\right)$
Simplify Fractions: Simplify each fraction by multiplying the numerators and denominators. $\newline$ First fraction becomes: $(\sqrt{5} - 2)(\sqrt{5} - 2) / ((\sqrt{5} + 2)(\sqrt{5} - 2))$ $\newline$ Second fraction becomes: $(\sqrt{5} + 2)(\sqrt{5} + 2) / ((\sqrt{5} - 2)(\sqrt{5} + 2))$
Expand Numerators: Now, let's expand the numerators. $\newline$ First fraction's numerator: $(\sqrt{5} - 2)(\sqrt{5} - 2) = 5 - 2\sqrt{5} - 2\sqrt{5} + 4$ $\newline$ Second fraction's numerator: $(\sqrt{5} + 2)(\sqrt{5} + 2) = 5 + 2\sqrt{5} + 2\sqrt{5} + 4$
Combine Like Terms: Combine like terms in the numerators. $\newline$ First fraction's numerator: $5 - 4\sqrt{5} + 4 = 9 - 4\sqrt{5}$ $\newline$ Second fraction's numerator: $5 + 4\sqrt{5} + 4 = 9 + 4\sqrt{5}$
Add Fractions: The denominators are the same and can be written as $(\sqrt{5} + 2)(\sqrt{5} - 2)$ , which simplifies to $5 - 4$ .
Simplify Denominator: Now, let's add the two fractions together. $\newline$ $(\frac{9 - 4\sqrt{5}}{5 - 4}) + (\frac{9 + 4\sqrt{5}}{5 - 4})$
Add Numerators: Simplify the denominator $5 - 4$ to get $1$ . $\frac{9 - 4\sqrt{5}}{1} + \frac{9 + 4\sqrt{5}}{1}$
Combine Like Terms: Add the numerators together since the denominators are both $1$ . $\newline$ $(9 - 4\sqrt{5}) + (9 + 4\sqrt{5})$
Cancel Terms: Combine like terms. $\newline$ $9 - 4\sqrt{5} + 9 + 4\sqrt{5}$
Add Remaining Terms: The terms $-4\sqrt{5}$ and $+4\sqrt{5}$ cancel each other out. $9 + 9$
Add Remaining Terms: The terms $-4\sqrt{5}$ and $+4\sqrt{5}$ cancel each other out. $9 + 9$ Add the remaining terms. $9 + 9 = 18$

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