## Euler L.'s A commentary on the continued fraction by which the PDF

By Euler L.

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Extra resources for A commentary on the continued fraction by which the illustrious La Grange has expressed the binomial powers

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Carry the base (v) from the dividend into the answer. There are no bases in the divisor that are not in the dividend, and vice versa. Next, subtract the exponent of v in the divisor from the exponent of v in the dividend: 9 – 5 = 4. The exponent of v in the answer is 4. (15v9)(3v5) = 5v4. 2. Begin with the coefﬁcients: –63 ÷ –7 = 9. Carry the base (a) from the dividend into the answer. There are no bases in the divisor that are not in the dividend, and vice versa. Next, subtract the exponent of a in the divisor from the exponent of a in the dividend: 1 – 8 = –7.

Replace g with –5: –8 + 2(–5) Multiply before adding: 2(–5) = –10 The expression becomes –8 + –10. Add: –8 + –10 = –18. Replace k with 20: 1 4(20) + 30 Multiply before adding: 1 4(20) = 5 The expression becomes 5 + 30. Add: 5 + 30 = 35. Replace x with 1: 12(4 – 1) Subtraction is in parentheses, so subtract before multiplying: (4 – 1) = 3 The expression becomes 12(3). Multiply: 12(3) = 36. Replace q with –6: (–6)2 + 15 Exponents come before addition in the order of operations, so handle the exponent ﬁrst: (–6)2 = 36 The expression becomes 36 + 15.

Remember, fractions mean division, so if a term appears in the denominator of a fraction, that means the term is acting like a divisor. Our answer, 7g 7g10y–4, could also be written as y . In fact, we could write both the 10 4 original problem and the answer as fractions: 35 g 10 4 5y 7 g 10 . 4 y = Now, it’s a little easier to see what happened: 35 was divided by 5, g10 was not divided by anything and went straight into our answer, and we could not divide by y4, so we kept it in the denominator of our answer.