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Find two irrational numbers with a product of i 6

Find two irrational numbers with a product of i 66

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Full solution

Q. Find two irrational numbers with a product of i 66
  1. Define irrational numbers: Let's call the two irrational numbers aa and bb. We know that a×b=i×6a \times b = i \times 6.
  2. Use imaginary unit: We know that ii is the imaginary unit, which is equal to the square root of −1-1. So, i×6i \times 6 is the same as 6i6i.
  3. Choose aa as 6\sqrt{6}: We can choose aa to be the square root of 66, which is an irrational number. So, a=6a = \sqrt{6}.
  4. Calculate bb: To find bb, we divide 6i6i by aa. So, b=6i6b = \frac{6i}{\sqrt{6}}.
  5. Rationalize denominator: Simplify bb by multiplying the numerator and denominator by 6\sqrt{6} to rationalize the denominator. b=6i×66×6b = \frac{6i \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}}.
  6. Simplify bb: After simplifying, b=66i6b = \frac{6\sqrt{6}i}{6}.
  7. Final result: Cancel out the 66 in the numerator and denominator. b=6ib = \sqrt{6}i.

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