Select the answer which is equivalent to the given expression using your calculator. $\newline$ $\frac{-8}{-20+\sqrt{18}}$ $\newline$ $\frac{80-4 \sqrt{18}}{191}$ $\newline$ $\frac{400-20 \sqrt{18}}{191}$ $\newline$ $\frac{80+4 \sqrt{18}}{191}$ $\newline$ $\frac{400+20 \sqrt{18}}{191}$

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Algebra 2

Roots of integers

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Q. Select the answer which is equivalent to the given expression using your calculator.

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\frac{-8}{-20+\sqrt{18}}

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\frac{80-4 \sqrt{18}}{191}

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\frac{400-20 \sqrt{18}}{191}

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\frac{80+4 \sqrt{18}}{191}

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\frac{400+20 \sqrt{18}}{191}

Simplify Square Root: Simplify the square root in the denominator. $\sqrt{18}$ can be simplified because $18 = 9 \times 2$ , and $9$ is a perfect square. $\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3 \times \sqrt{2}$
Substitute Simplified Expression: Substitute the simplified square root back into the expression. $\newline$ The original expression $(-8)/(-20+\sqrt{18})$ becomes $(-8)/(-20+3\sqrt{2})$
Match Simplified Form: Look for the expression that matches the simplified form. $\newline$ We need to find which of the given options matches $(-8)/(-20+3\sqrt{2})$ when simplified.
Check Options for Match: Simplify the given options to see if any match the expression from Step $2$ . $\newline$ Option $1$ : $(80-4\sqrt{18})/(191)$ simplifies to $(80-4\cdot 3\cdot\sqrt{2})/(191) = (80-12\sqrt{2})/(191)$ $\newline$ Option $2$ : $(400-20\sqrt{18})/(191)$ simplifies to $(400-20\cdot 3\cdot\sqrt{2})/(191) = (400-60\sqrt{2})/(191)$ $\newline$ Option $3$ : $(80+4\sqrt{18})/(191)$ simplifies to $(80+4\cdot 3\cdot\sqrt{2})/(191) = (80+12\sqrt{2})/(191)$ $\newline$ Option $4$ : $(400+20\sqrt{18})/(191)$ simplifies to $(400+20\cdot 3\cdot\sqrt{2})/(191) = (400+60\sqrt{2})/(191)$ $\newline$ None of these options match $(-8)/(-20+3\cdot\sqrt{2})$ directly. However, we can multiply the numerator and denominator of our expression by $-1$ to get a positive denominator and see if it matches any option: $\newline$ $(80-4\cdot 3\cdot\sqrt{2})/(191) = (80-12\sqrt{2})/(191)$ $0$
Re-evaluate Simplification: Check if the expression $(8)/(20-3\sqrt{2})$ matches any of the options. $\newline$ Option $1$ : $(80-12\sqrt{2})/(191)$ does not match because the numerator is not $8$ . $\newline$ Option $2$ : $(400-60\sqrt{2})/(191)$ does not match because the numerator is not $8$ . $\newline$ Option $3$ : $(80+12\sqrt{2})/(191)$ does not match because the sign in front of the square root is positive. $\newline$ Option $4$ : $(400+60\sqrt{2})/(191)$ does not match because the numerator is not $8$ . $\newline$ None of the options match the expression $(8)/(20-3\sqrt{2})$ . This means there might be a mistake in the simplification or the given options do not include the correct equivalent expression.
Re-evaluate Simplification: Check if the expression $(8)/(20-3\sqrt{2})$ matches any of the options. $\newline$ Option $1$ : $(80-12\sqrt{2})/(191)$ does not match because the numerator is not $8$ . $\newline$ Option $2$ : $(400-60\sqrt{2})/(191)$ does not match because the numerator is not $8$ . $\newline$ Option $3$ : $(80+12\sqrt{2})/(191)$ does not match because the sign in front of the square root is positive. $\newline$ Option $4$ : $(400+60\sqrt{2})/(191)$ does not match because the numerator is not $8$ . $\newline$ None of the options match the expression $(8)/(20-3\sqrt{2})$ . This means there might be a mistake in the simplification or the given options do not include the correct equivalent expression.Re-evaluate the simplification and the given options. $\newline$ Upon re-evaluation, it seems that there was a mistake in the previous steps. The expression $(-8)/(-20+\sqrt{18})$ should be equivalent to one of the given options. Let's multiply the numerator and denominator of our expression by the conjugate of the denominator to rationalize it: $\newline$ $(80-12\sqrt{2})/(191)$ $0$