Lesson 8 · from §93 of the 1910 text
Addition and Subtraction of Radicals
Addition and subtraction of radicals
Similar radicals are radicals of the same order whose radicands are identical, or can be made so by simplification. Thus 3\sqrt[3]{ac^2} and \sqrt[3]{ac^2} are similar, and so are \sqrt8 and \sqrt{18}, for \sqrt8 = 2\sqrt2 and \sqrt{18} = 3\sqrt2. Dissimilar radicals are radicals that are not similar.
The sum or difference of similar radicals can be expressed as one term; that of dissimilar radicals can only be indicated: 5\sqrt2 + 3\sqrt2 = 8\sqrt2, \qquad\text{but}\qquad 5\sqrt2 + 3\sqrt[3]{2} \text{ stays } 5\sqrt2 + 3\sqrt[3]2.
Example 1. \sqrt{12} + \sqrt{48} - \sqrt{75} = 2\sqrt3 + 4\sqrt3 - 5\sqrt3 = \sqrt3.
Example 2. \sqrt[3]{24a^3} + 2\sqrt[3]{81a^3} - \sqrt[3]{3a^3} = 2a\sqrt[3]3 + 6a\sqrt[3]3 - a\sqrt[3]3 = 7a\sqrt[3]3.
Example 3. \sqrt{\tfrac12} + \sqrt{18} - \sqrt{\tfrac13} = \tfrac12\sqrt2 + 3\sqrt2 - \tfrac13\sqrt3 = 3\tfrac12\sqrt2 - \tfrac13\sqrt3.
So: simplify each radical, then collect the similar ones. When the coefficients contain letters, collect them in a parenthesis: a\sqrt x + 2\sqrt x = (a + 2)\sqrt x.
How to type radicals
- Square root:
sqrt(12)or√12; a coefficient goes in front:2sqrt(3)or2√3. - Cube root:
cbrt(5)or∛5. Any other root:root(x, 4)for \sqrt[4]{x},root(2, 6)for \sqrt[6]{2}. - Put the whole radicand in parentheses when it is more than one number or letter:
sqrt(2x),cbrt(a + b),sqrt(2 - sqrt(2)). - Fractional exponents use
^with parentheses:x^(2/3). - Letters under a radical sign stand for positive numbers, so \sqrt{x^2} = x.
The checker also wants the similar radicals collected: 2\sqrt3 + 4\sqrt3 should be written 6\sqrt3.
From the history of algebra
Though methods of classifying irrational expressions are found in the works of Euclid, the Hindus and the Arabs were the first to develop this part of algebra in anything like the form used to-day. The word surd comes from a mistranslation of a Greek word meaning not absurd or foolish but inexpressible, that is, inexpressible in terms of rational numbers.
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.
Simplify and collect:
\sqrt{27} + \sqrt{12}
\sqrt{45} - \sqrt{20}
2\sqrt{200} - 3\sqrt8
\sqrt[3]{56} + 2\sqrt[3]{189}
2\sqrt[3]{320} - \sqrt[2]{50}
\sqrt[4]{25} - \sqrt[2]{50}
\sqrt[4]{32} + 5\sqrt[4]{162}
5\sqrt{\tfrac12} - \tfrac32\sqrt2
\sqrt{\tfrac13} + 2\sqrt{\tfrac43} - 3\sqrt{\tfrac23}
\sqrt{\tfrac3{10}} - \sqrt{120} - 2\sqrt{\tfrac65}
\sqrt{\tfrac5{18}} + 2\sqrt{\tfrac{32}5} - \sqrt{\tfrac{10}9}
\sqrt[3]{\tfrac68} - \sqrt[3]{12} + \sqrt[3]6
a\sqrt[4]4 + \sqrt{8a^2}
R - \sqrt{\dfrac{3R^2}{4}}
2x\sqrt[3]{54x} - 3\sqrt[3]{16x^4} + \sqrt[6]{4x^2}
\sqrt[3]{81x^7} + x\sqrt[3]{375x^4} - \sqrt[12]{16x^4}
\sqrt{a^3bc} - a\sqrt{abc} + ac\sqrt{\dfrac{b}{ac}}
\sqrt{(m + n)^3} - n\sqrt[4]{(m + n)^2}
\sqrt[3]{(a + b)^4} - \sqrt[3]{8a^3(a + b)} + \sqrt[6]{a^2 + 2ab + b^2}
\sqrt[2]{a^3 + 4a^2 + 4a} - \sqrt[2]{a^3} - \dfrac{2}{a^3}\sqrt{a^7}
\sqrt{x^3y^3} - x^2y^2\sqrt{\dfrac{1}{xy}} + xy\sqrt{2 + \dfrac{x^2 + y^2}{xy}}
rs\sqrt[3]{rs} + \sqrt[3]{\dfrac{1}{r^2s^2}} - 2\sqrt[3]{r^4s^4}
2\sqrt{\dfrac yx} - \sqrt{\dfrac xy} + \sqrt{2 + \dfrac{x^2 + y^2}{xy}}
\sqrt{3x^2 - 18x + 27} - \sqrt{27(x^2 + 2x + 1)}
\sqrt[3]{(a - 3)^2(5a - 15)} + \sqrt[3]{40}