Full solution

Q. Simplify. Rationalize the denominator.

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\frac{8}{2 - \sqrt{2}}

Select Conjugate: Select the conjugate of $2 - \sqrt{2}$ . Conjugate of a number in the form $a - \sqrt{b}$ is $a + \sqrt{b}$ . So, the conjugate of $2 - \sqrt{2}$ is $2 + \sqrt{2}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. $\newline$ We have the expression $\frac{8}{2 - \sqrt{2}}$ . To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is $2 + \sqrt{2}$ . $\newline$ So, $\frac{8 \times (2 + \sqrt{2})}{(2 - \sqrt{2}) \times (2 + \sqrt{2})}$
Simplify Numerator: Simplify the numerator. $\newline$ Now we multiply $8$ by each term in the conjugate $2 + \sqrt{2}$ . $\newline$ $8 \times 2 + 8 \times \sqrt{2}$ $\newline$ $= 16 + 8 \times \sqrt{2}$
Simplify Denominator: Simplify the denominator using the difference of squares formula. $\newline$ The denominator $(2 - \sqrt{2}) * (2 + \sqrt{2})$ simplifies to $2^2 - (\sqrt{2})^2$ . $\newline$ $2^2 - (\sqrt{2})^2$ $\newline$ = $4 - 2$ $\newline$ = $2$
Write Simplified Expression: Write the simplified expression. $\newline$ Now we have the simplified numerator and denominator. $\newline$ So, the expression becomes $(16 + 8 \cdot \sqrt{2})/2$ .
Divide Numerator by Denominator: Simplify the expression by dividing the numerator by the denominator. $\newline$ Since both terms in the numerator are divisible by $2$ , we divide them by $2$ . $\newline$ $(16/2) + (8 \times \sqrt{2})/2$ $\newline$ = $8 + 4 \times \sqrt{2}$