Lesson 5 · from §110 of the 1910 text
Division with Any Exponents
Dividing with exponents
Dividing. A single term divides by subtracting exponents: ax^{\frac23} \div a^{\frac12}x^{\frac12} = a^{\frac12}x^{\frac16}. To divide by a polynomial, arrange both dividend and divisor in descending powers of a letter, counting fractional and negative exponents in their order, and divide as in long division.
Example 1. Divide x - y by x^{\frac13} - y^{\frac13}.
Think of x as (x^{\frac13})^3 and y as (y^{\frac13})^3; then this is the familiar (t^3 - u^3) \div (t - u) = t^2 + tu + u^2, so the quotient is
x^{\frac23} + x^{\frac13}y^{\frac13} + y^{\frac23}.
Example 2. Divide x^{2a} - 2 + x^{-2a} by x^a - x^{-a}. Here x^{2a} - 2 + x^{-2a} = (x^a - x^{-a})^2, so the quotient is x^a - x^{-a}.
Check every quotient by multiplying it by the divisor.
Quick check
Divide x - y by x^{\frac13} - y^{\frac13}.
How to type exponents
- A power:
x^3. A negative or fractional exponent goes in parentheses:x^(-2),x^(2/3),a^(-1/2). (You may writex^-2, butx^-1/2means \frac{x^{-1}}{2}, so usex^(-1/2).) - An exponent with letters:
e^(2x),x^(a+1),m^(n-1). - Radicals are accepted too:
sqrt(x),cbrt(x),root(x, 4).
From the history of algebra
John Wallis. To us, who use the notation of exponents every day, it seems so simple and natural a method of expressing the product of several equal factors that it is difficult to understand why such a long time was necessary to develop it. But here, as in many other instances, it required a great man to discover what to us seems the most obvious relation. The man who brought the notation of exponents to its modern form was John Wallis (1616–1703), an Englishman. He was the son of a clergyman, and, like most scholars of his day, did not confine his interests to any one subject. Wallis became widely known by deciphering a military dispatch which contained a hidden meaning, and all his life was interested in such puzzling problems. He was at one time an instructor in Latin, Greek, and Hebrew, wrote books on theology and English grammar, and invented a method of teaching deaf mutes to talk. He was the most notable English mathematician before Sir Isaac Newton, who highly prized Wallis's work.
Though the idea of using negative and fractional exponents had occurred to writers before Wallis, it was he who showed their naturalness, and who introduced them permanently. He also was the first to use the ordinary sign \infty to denote infinity.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Perform the indicated division:
x^4 \div x^6
x^8 \div x^{\frac13}
x^{\frac12} \div x^2
ax^{\frac23} \div a^{\frac12}x^{\frac12}
\dfrac{ax - a^2x^3}{a^2x^{\frac12}}
\left(x^a - 2x^{2a-1} + 3x^{3a-2}\right) \div x^{2a-1}
\left(6a^{3+4n} - 9a^{n-2} + 5a^{2-n}\right) \div 3a^{n-2}
(x - y) \div \left(x^{\frac12} - y^{\frac12}\right)
(x + y) \div \left(x^{\frac13} + y^{\frac13}\right)
(x - 8y) \div \left(x^{\frac13} - 2y^{\frac13}\right)
(16x^2 - 81y^2) \div \left(2x^{\frac12} + 3y^{\frac12}\right)
(a^3 - b^2) \div \left(\sqrt a + \sqrt[3]{b}\right)
\left(a^2 + ab^{-1} + b^{-2}\right) \div \left(a - a^{\frac12}b^{-\frac12} + b^{-1}\right)
\left(e^{2x} - 2 + e^{-2x}\right) \div (e^x - e^{-x})
\left(e^{3x} + 3e^x + \dfrac{3}{e^x} + \dfrac{1}{e^{3x}}\right) \div (e^x + e^{-x})
\left(e^{4x} - 4e^{2x} + 6 - 4e^{-2x} + e^{-4x}\right) \div (e^x - e^{-x})
\left(a + a^{\frac35}b^{\frac23} + 2a^{\frac25}b^{\frac13} + 2b\right) \div \left(a^{\frac35} + 2b^{\frac13}\right)
\left(m^4 + 7m^2 + 8 - 7m^{-2} + m^{-4}\right) \div \left(m^2 + 5 - m^{-2}\right)
\left(2x^{2n-1} - x^{3n-2} + 9x^{5n-4}\right) \div \left(2x^{n-1} + 3x^{2n-2}\right)
\left(x^{-\frac52}y - 2x^{-\frac32}y^{\frac12} - 8y^{\frac12} + 16x\right) \div \left(x^{-\frac12}y^{-\frac12} - 8x^2y^{-1}\right)
\left(40ab - 16a^{\frac{19}{5}}b^{\frac34} - 25a^{-\frac95}b^{\frac54}\right) \div \left(5a^{-\frac25}b^{-\frac38} - 4a^{\frac{12}{5}}b^{-\frac58}\right)
\left(2x^{-2a} - 28x^{-a} + 33 + 38x^a + 11x^{2a} + x^{3a}\right) \div \left(x^a - 2x^{-a} + 4\right)
\left(25m^3 - 4mn^{\frac43} + 4mn^2 - mn^{\frac83}\right) \div \left(5m^{\frac32} - 2n^{\frac23}m^{\frac12} + m^{\frac12}n^{\frac43}\right)