Final answer:
Safiya can select values A (3^12), B (3^4), C (3^3), D (3^1/18), and E (3^18) to multiply with 3^18 to achieve a rational product, as these combinations result in integer exponents when their powers are added together.
Step-by-step explanation:
Safiya is seeking a number that can be multiplied by 3^18 to result in a rational product. In order to determine which values among the given options would achieve this, we need to add the exponents of the like base when multiplying powers of the same base. If the result has an integer exponent, then the product is rational.
- A. 3^12 multiplied by 3^18 equals 3^(12+18) which simplifies to 3^30. This is a rational number since it has an integer exponent.
- B. 3^4 multiplied by 3^18 equals 3^(4+18), which simplifies to 3^22, also a rational number.
- C. 3^3 multiplied by 3^18 equals 3^(3+18), which simplifies to 3^21, again a rational number.
- D. 3^(1/18) multiplied by 3^18 equals 3^((1/18)+18), which simplifies to 3^(18/18+18), i.e., 3^1, also a rational number.
- E. 3^18 multiplied by 3^18 equals 3^(18+18), which simplifies to 3^36, a rational number.
- F. 3^(1/3) multiplied by 3^18 equals 3^((1/3)+18), which simplifies to 3^(54/3+1/3), i.e., 3^55/3, which is not a rational number because the exponent is not an integer.
- G. 3^(1/6) multiplied by 3^18 equals 3^((1/6)+18), which simplifies to 3^(108/6+1/6), i.e., 3^109/6, not a rational number for the same reason as F.
So the values that Safiya could use to result in a rational product when multiplied by 3^18 are options A, B, C, D, and E.