Final answer:
The repeating decimal .428571429 can be converted into the fraction 3/7 through algebraic manipulation that involves multiplying by a power of 10 and subtracting to isolate the repeating pattern.
Step-by-step explanation:
The number .428571429 can be expressed as a fraction by recognizing that it is a repeating decimal. To convert a repeating decimal into a fraction, we can use algebraic methods. Let x equal the repeating decimal .428571429. To isolate the repeating portion, multiply x by 10^6 (since there are 6 repeating digits), to get 10^6x. Subtracting x from 10^6x we get 10^6x - x = 428571.429 - .428571429. Simplifying, 999999x = 428571. Solving for x we find that x = 428571/999999, which can be simplified to 3/7 after recognizing the repeating pattern relates to this fraction.