Select the answer which is equivalent to the given expression using your calculator. (-8)/(-20+sqrt18) (80-4sqrt18)/(191) (400-20sqrt18)/(191) (80+4sqrt18)/(191) (400+20sqrt18)/(191)

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Select the answer which is equivalent to the given expression using your calculator.  (-8)/(-20+sqrt18)  (80-4sqrt18)/(191)  (400-20sqrt18)/(191)  (80+4sqrt18)/(191)  (400+20sqrt18)/(191)

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Simplify Square Root: Simplify the square root in the denominator. sqrt(18) can be simplified because 18 = 9 * 2, and 9 is a perfect square. sqrt(18) = sqrt(9 * 2) = sqrt(9) * sqrt(2) = 3 * sqrt(2)Substitute Simplified Expression: Substitute the simplified square root back into the expression. The original expression (-8)/(-20+sqrt18) becomes (-8)/(-20+3*sqrt(2))Match Simplified Form: Look for the expression that matches the simplified form. 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. Option 1: (80-4sqrt18)/(191) simplifies to (80-4*3*sqrt(2))/(191) = (80-12sqrt(2))/(191) Option 2: (400-20sqrt18)/(191) simplifies to (400-20*3*sqrt(2))/(191) = (400-60sqrt(2))/(191) Option 3: (80+4sqrt18)/(191) simplifies to (80+4*3*sqrt(2))/(191) = (80+12sqrt(2))/(191) Option 4: (400+20sqrt18)/(191) simplifies to (400+20*3*sqrt(2))/(191) = (400+60sqrt(2))/(191)  None of these options match (-8)/(-20+3*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: (-8)/(-20+3*sqrt(2)) * (-1/-1) = (8)/(20-3*sqrt(2))Re-evaluate Simplification: Check if the expression (8)/(20-3*sqrt(2)) matches any of the options. Option 1: (80-12sqrt(2))/(191) does not match because the numerator is not 8. Option 2: (400-60sqrt(2))/(191) does not match because the numerator is not 8. Option 3: (80+12sqrt(2))/(191) does not match because the sign in front of the square root is positive. Option 4: (400+60sqrt(2))/(191) does not match because the numerator is not 8.  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. Upon re-evaluation, it seems that there was a mistake in the previous steps. The expression (-8)/(-20+sqrt18) 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: (-8)/(-20+3*sqrt(2)) * (-20-3*sqrt(2))/(-20-3*sqrt(2)) = (-8 * -20 - 8 * -3*sqrt(2))/((-20)^2 - (3*sqrt(2))^2)

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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