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$e) A fraction with 2 mixed radicals in the numerator and 2 mixed radicals in the denominator whose result contains a perfect square in the numerator when rationalized.$

e) A fraction with $2$ mixed radicals in the numerator and $2$ mixed radicals in the denominator whose result contains a perfect square in the numerator when rationalized.

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Q. e) A fraction with

2

mixed radicals in the numerator and

2

mixed radicals in the denominator whose result contains a perfect square in the numerator when rationalized.

Pick Fraction with Mixed Radicals: Let's pick a fraction with mixed radicals. How about something like $(\sqrt{2} + \sqrt{3}) / (\sqrt{5} + \sqrt{7})$ ? We want to rationalize this.
Rationalize Denominator: To rationalize the denominator, we'll multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $(\sqrt{5} + \sqrt{7})$ is $(\sqrt{5} - \sqrt{7})$ .
Distribute and Simplify Numerator: So we multiply $(\sqrt{2} + \sqrt{3}) / (\sqrt{5} + \sqrt{7})$ by $(\sqrt{5} - \sqrt{7}) / (\sqrt{5} - \sqrt{7})$ .
Use Difference of Squares for Denominator: In the numerator, we'll use the distributive property: $(\sqrt{2} + \sqrt{3}) \times (\sqrt{5} - \sqrt{7}) = \sqrt{2}\times\sqrt{5} - \sqrt{2}\times\sqrt{7} + \sqrt{3}\times\sqrt{5} - \sqrt{3}\times\sqrt{7}$ .
Finalize Numerator with Perfect Square: Simplify the numerator: $\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}$ .
Choose Fraction with Perfect Squares: In the denominator, we'll use the difference of squares: $(\sqrt{5} + \sqrt{7}) \times (\sqrt{5} - \sqrt{7}) = 5 - 7$ .
Simplify Fraction with Perfect Squares: Simplify the denominator: $5 - 7 = -2$ .
Simplify Fraction with Perfect Squares: Simplify the denominator: $5 - 7 = -2$ .Now we have $(\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}) / -2$ . But we want a perfect square in the numerator.
Simplify Fraction with Perfect Squares: Simplify the denominator: $5 - 7 = -2$ .Now we have $\frac{\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}}{-2}$ . But we want a perfect square in the numerator.Let's try a different approach. We'll start with a fraction that has a perfect square when multiplied out. How about $\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}}$ , where $a$ and $b$ are perfect squares?
Simplify Fraction with Perfect Squares: Simplify the denominator: $5 - 7 = -2$ .Now we have $(\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}) / -2$ . But we want a perfect square in the numerator.Let's try a different approach. We'll start with a fraction that has a perfect square when multiplied out. How about $(\sqrt{a} + \sqrt{b}) / (\sqrt{a} - \sqrt{b})$ , where $a$ and $b$ are perfect squares?Let's pick $a = 4$ and $b = 1$ , so we have $(\sqrt{4} + \sqrt{1}) / (\sqrt{4} - \sqrt{1})$ .
Simplify Fraction with Perfect Squares: Simplify the denominator: $5 - 7 = -2$ .Now we have $(\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}) / -2$ . But we want a perfect square in the numerator.Let's try a different approach. We'll start with a fraction that has a perfect square when multiplied out. How about $(\sqrt{a} + \sqrt{b}) / (\sqrt{a} - \sqrt{b})$ , where $a$ and $b$ are perfect squares?Let's pick $a = 4$ and $b = 1$ , so we have $(\sqrt{4} + \sqrt{1}) / (\sqrt{4} - \sqrt{1})$ .Simplify the fraction: $(2 + 1) / (2 - 1)$ .
Simplify Fraction with Perfect Squares: Simplify the denominator: $5 - 7 = -2$ .Now we have $(\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}) / -2$ . But we want a perfect square in the numerator.Let's try a different approach. We'll start with a fraction that has a perfect square when multiplied out. How about $(\sqrt{a} + \sqrt{b}) / (\sqrt{a} - \sqrt{b})$ , where $a$ and $b$ are perfect squares?Let's pick $a = 4$ and $b = 1$ , so we have $(\sqrt{4} + \sqrt{1}) / (\sqrt{4} - \sqrt{1})$ .Simplify the fraction: $(2 + 1) / (2 - 1)$ .Now we have $3 / 1$ , which is just $(\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}) / -2$ $0$ . But that's not a fraction with mixed radicals in the numerator and denominator.