Final answer:
The product of 0.2 recurring (written as 2/9) and 0.136 with 36 recurring (written as 136/999) is 272/8991. This can be approximated by performing the division of 272 by 8991 to get the decimal representation.
Step-by-step explanation:
The student's question is about finding the product of a recurring decimal, 0.2 recurring, and another number with recurring last digits, 0.136 with 36 recurring. To tackle this, we first need to express the recurring decimals in fractional form. The recurring decimal 0.2 recurring can be written as 2/9, since 0.222... is equivalent to the infinite series 2/10 + 2/100 + 2/1000 + ..., which sums up to 2/9.
For the decimal 0.136 with 36 recurring, we can write it as 136/999, because the 36 at the end is the recurring part. Now, we multiply these two fractions together. Performing the multiplication, (2/9) × (136/999), we get 272/8991, which is the answer in fractional form. To get the decimal form, we can divide 272 by 8991, which would yield the approximate decimal representation of the product.