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Simplify
(6-4sqrt3)/(6+4sqrt3) by rationalizing the denominator.
If
x=3+2sqrt2, then find whether
x+(1)/(x) is rational or irrational

$18$ . Simplify $\frac{6-4 \sqrt{3}}{6+4 \sqrt{3}}$ by rationalizing the denominator. $\newline$ $6$ . If $x=3+2 \sqrt{2}$ , then find whether $x+\frac{1}{x}$ is rational or irrational

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Transformations of linear functions

Full solution

Q.

18

. Simplify

\frac{6-4 \sqrt{3}}{6+4 \sqrt{3}}

by rationalizing the denominator.

\newline

6

. If

x=3+2 \sqrt{2}

, then find whether

x+\frac{1}{x}

is rational or irrational

Multiply by conjugate: To simplify $(6-4\sqrt{3})/(6+4\sqrt{3})$ , we'll multiply the numerator and denominator by the conjugate of the denominator. $\newline$ Calculation: $(6-4\sqrt{3})/(6+4\sqrt{3}) \times (6-4\sqrt{3})/(6-4\sqrt{3})$
Expand numerator and denominator: Now, we'll expand the numerator and the denominator. $\newline$ Calculation: $(6-4\sqrt{3})^2 / (6+4\sqrt{3})(6-4\sqrt{3})$
Simplify squares and product: Simplify the squares and the product of the conjugates. $\newline$ Calculation: $(36-48\sqrt{3}+48)/(36-48)$
Combine like terms: Combine like terms in the numerator and denominator. $\newline$ Calculation: $(84-48\sqrt{3})/(-12)$
Divide by denominator: Divide both terms in the numerator by the denominator. $\newline$ Calculation: $-7+4\sqrt{3}$

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