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    Are there any shortcuts we could take on this question?

    Module 4 Day 6 Challenge Part 2
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    • The Blade DancerT
      The Blade Dancer M0★ M1★ M2★ M3★ M4 M5
      last edited by

      I understood this mini-question explanation but it's rather tedious lol. Are there any shortcuts I could use in a situation such as a timed contest?

      The Blade Dancer
      League of Legends, Valorant: Harlem Charades (#NA1)
      Discord: Change nickname if gay#7585

      quacker88Q 1 Reply Last reply Reply Quote 1
      • quacker88Q
        quacker88 MOD @The Blade Dancer
        last edited by quacker88

        Great question, @the-blade-dancer !

        If you remember the binomial theorem at all from Module 3 (let me know if you need more explanation here), but we can use a shortcut for the expansion of \((\sqrt{a}+\sqrt{b})^3\).

        In general, \((x+y)^3=x^3+3x^2y+3xy^2+y^3\), so

        \((\sqrt{a}+\sqrt{b})^3=a\sqrt a +3a\sqrt b+ 3\sqrt a b+b\sqrt b\)
        \((\sqrt{a}+\sqrt{b})^3= (a+3b)\sqrt a+(3a+b)\sqrt b\)

        This gets us straight to the form that we need. And since this has to be equal to \(22\sqrt 7+26\sqrt 5\), it is most likely that \(a=7, b=5\), so we can go and check that out. After confirming that it is true, we get that our answer is \(7+5=\boxed{12}\). Let me know if you have any other questions!

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