Lesson 14 · from §95 of the 1910 text
Binomial Divisors and Approximate Values
Binomial divisors
When the divisor is a binomial (or polynomial) containing radicals, the practical method of division is indirect: multiply both terms of the fraction by a rationalizing factor of the divisor.
Example 7. 8 \div \left(3 + \sqrt7\right) = \dfrac{8\left(3 - \sqrt7\right)}{\left(3 + \sqrt7\right)\left(3 - \sqrt7\right)} = \dfrac{24 - 8\sqrt7}{9 - 7} = 12 - 4\sqrt7.
Example 8. \left(\sqrt5 + \sqrt3\right) \div \left(2\sqrt5 - \sqrt3\right) = \dfrac{\left(\sqrt5 + \sqrt3\right)\left(2\sqrt5 + \sqrt3\right)}{\left(2\sqrt5 - \sqrt3\right)\left(2\sqrt5 + \sqrt3\right)} = \dfrac{10 + 3\sqrt{15} + 3}{20 - 3} = \dfrac{13}{17} + \dfrac{3}{17}\sqrt{15}.
Rule
Write the dividend over the divisor in the form of a fraction. Then multiply the numerator and denominator of the fraction by the rationalizing factor of the denominator and simplify the resulting fraction.
The same method changes a fraction with letters to one with a rational denominator: \dfrac{\sqrt x}{\sqrt x - \sqrt y} = \dfrac{\sqrt x\left(\sqrt x + \sqrt y\right)}{x - y} = \dfrac{x + \sqrt{xy}}{x - y}.
Approximate values
Rationalize first, then use the values \sqrt2 = 1.4142, \sqrt3 = 1.7321, \sqrt5 = 2.2361, \sqrt6 = 2.4495, \sqrt7 = 2.6458: \frac{1}{\sqrt5 - 2} = \sqrt5 + 2 = 4.236. Every irrational expression made of rational numbers and radicals has a rationalizing factor, but finding it can be far beyond us: for \dfrac{\sqrt2 + \sqrt3}{\sqrt2 + \sqrt[3]2 + \sqrt[4]2}, just divide the sum of the approximate roots above the line by the sum of those below.
Typing answers: exact answers as in the last lesson, like
12 - 4sqrt(7)or(x + sqrt(x*y))/(x - y). Approximate values to three decimals, like4.236(within about 0.002 is accepted).
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.
5 \div \left(\sqrt5 + 2\right)
\dfrac{2}{2 - \sqrt3}
4 \div \left(\sqrt3 - \sqrt2\right)
\sqrt5 \div \left(3\sqrt2 - \sqrt5\right)
\left(\sqrt5 - \sqrt3\right) \div \left(\sqrt5 + \sqrt3\right)
\dfrac{4\sqrt3 + 2\sqrt2}{4\sqrt3 - 2\sqrt2}
\dfrac{2\sqrt5 - 3\sqrt3}{5\sqrt5 - 5}
Find to three decimals the approximate values of the following:
3 + \sqrt2
14 - 5\sqrt7
6 \pm 2\sqrt5
\dfrac{7 \pm \sqrt6}{3}
3\sqrt6 \div 2\sqrt5
\dfrac{2\sqrt5 + 1}{3\sqrt5 - \sqrt3}
\sqrt{2 - \sqrt3}
\dfrac{\sqrt3}{\sqrt3 + \sqrt[3]4 + \sqrt[4]5}, given \sqrt[3]4 = 1.5874.
Change the following fractions to equivalent fractions having rational denominators:
\dfrac{\sqrt a + \sqrt b}{\sqrt a - \sqrt b}
\dfrac{2\sqrt x - \sqrt a}{\sqrt x + 3\sqrt a}
\dfrac{r\sqrt3 + \sqrt r}{\sqrt3 - \sqrt r}
\dfrac{m\sqrt n + a\sqrt b}{m\sqrt n - a\sqrt b}
\dfrac{\sqrt{a - 2} - 2}{\sqrt{a - 2} + 2}
\dfrac{2}{\sqrt[4]2 + \sqrt2}
Perform the indicated division:
\left(x - \sqrt{a + b}\right) \div \left(x + \sqrt{a + b}\right)
\left(\sqrt a + \sqrt b\right) \div \left(\sqrt c + \sqrt d\right)
Is there any real distinction between the direction which precedes Exercise 48 and that which precedes Exercise 54?