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a Use a calculator to show that sqrt2×sqrt32 is a rational b Find two irrational numbers with a product of

a Use a calculator to show that 2×32 \sqrt{2} \times \sqrt{32} is a rational\newlineb Find two irrational numbers with a product of

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Full solution

Q. a Use a calculator to show that 2×32 \sqrt{2} \times \sqrt{32} is a rational\newlineb Find two irrational numbers with a product of
  1. Calculate product of square roots: Use a calculator to find the product of 2×32\sqrt{2}\times\sqrt{32}.\newlineCalculation: 2×32=64=8\sqrt{2}\times\sqrt{32} = \sqrt{64} = 8.
  2. Check for rational number: Check if the result is a rational number.\newlineRational numbers can be expressed as a fraction of two integers. Since 88 is an integer, it can be written as 81\frac{8}{1}, which is a fraction of two integers.
  3. Find irrational numbers for rational product: Find two irrational numbers that multiply to a rational number.\newlineLet's take the square root of 22 (2\sqrt{2}) and the square root of 88 (8\sqrt{8}). Both are irrational.\newlineCalculation: 2×8=16=4\sqrt{2}\times\sqrt{8} = \sqrt{16} = 4.

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