Full solution

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

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\frac{-13}{-1-\sqrt{14}}

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-7+7 \sqrt{14}

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-1+\sqrt{14}

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-7-7 \sqrt{14}

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-1-\sqrt{14}

Simplify Denominator: Simplify the denominator of the given expression. $\newline$ We have the expression $(-13)/(-1-\sqrt{14})$ . To simplify, we first look at the denominator, which is $-1-\sqrt{14}$ . We can rewrite this as $-(1+\sqrt{14})$ to make it clearer that the entire expression is negative.
Divide Numerator: Divide the numerator by the simplified denominator. $\newline$ Now we divide $-13$ by $-(1+\sqrt{14})$ . Since we have a negative divided by a negative, the result will be positive. So, the expression simplifies to $\frac{13}{1+\sqrt{14}}$ .
Rationalize Denominator: Rationalize the denominator. $\newline$ To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $1+\sqrt{14}$ is $1-\sqrt{14}$ . So we multiply both the numerator and the denominator by $1-\sqrt{14}$ .
Perform Multiplication: Perform the multiplication. $\newline$ Multiplying the numerator: $13 \times (1-\sqrt{14}) = 13 - 13\sqrt{14}$ . $\newline$ Multiplying the denominator: $(1+\sqrt{14}) \times (1-\sqrt{14}) = 1 - (\sqrt{14})^2$ .
Evaluate Squares: Evaluate the squares and simplify the denominator. $(\sqrt{14})^2$ is simply $14$ . So the denominator becomes $1 - 14$ , which is $-13$ .
Simplify Entire Expression: Simplify the entire expression. $\newline$ Now we have $(13 - 13\sqrt{14}) / -13$ . Both terms in the numerator can be divided by $-13$ , which gives us $-1 + \sqrt{14}$ .