Lesson 1 · from §47, 48 of the 1910 text
Roots of Monomials
Factoring means finding two or more expressions whose product is a given expression. It's a big subject, so this chapter sticks to the common forms, and only to factors without irrational numbers (like \sqrt{2}) or fractions. So x^2 - 2y^2 counts as prime here, even though (x + \sqrt2\,y)(x - \sqrt2\,y) = x^2 - 2y^2.
Square roots
Factoring often needs the roots of monomials. The square root of a monomial is one of the two equal factors whose product is the monomial. Since (+2)(+2) = 4 and (-2)(-2) = 4, the square root of 4 is \pm 2.
Every positive number has two square roots with the same absolute value and opposite signs.
The positive one is the principal square root, and a bare radical means that one: \sqrt{4} = 2, and -\sqrt4 = -2. Since a^5 \cdot a^5 = a^{10}, \sqrt{a^{10}} = a^5.
Rule. Take the square root of the coefficient, then give each letter half its exponent.
The same idea works for fourth and sixth roots: \sqrt[4]{81a^4} = 3a, \sqrt[6]{x^{12}} = x^2. A negative term like -4 or -9a^2 has no square root here, since two equal factors can't multiply to a negative.
Quick check
\sqrt{36a^8b^2}
Cube roots
The cube root is one of three equal factors. 3 \cdot 3 \cdot 3 = 27, so \sqrt[3]{27} = 3; and (-3)^3 = -27, so \sqrt[3]{-27} = -3.
The cube root of a monomial has the same sign as the monomial. Give each letter one third of its exponent.
Likewise for fifth and seventh roots: \sqrt[5]{x^{10}} = x^2, \sqrt[7]{128x^{21}} = 2x^3.
Quick check
\sqrt[3]{-64x^6}
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.
Find the principal root.
\sqrt{4a^2}
\sqrt{9a^4}
\sqrt{25a^6x^4}
\sqrt{4x^4}
\sqrt{9x^6}
\sqrt{16x^{16}}
\sqrt{49x^{14}y^{10}}
\sqrt{81x^{12}d^{20}}
\sqrt{121c^{22}d^6}
\sqrt{169x^6}
\sqrt{196y^{10}}
\sqrt{225y^4z^6}
\sqrt{256c^{12}}
\sqrt{400a^2b^2c^4}
\sqrt{324b^8}
\sqrt{625b^2y^{18}z^6}
\sqrt{x^{2n}}
\sqrt{x^{4n}}
\sqrt{x^{6n}}
\sqrt{a^{4n}b^{8n}}
\sqrt[3]{8}
\sqrt[3]{-8a^3}
\sqrt[3]{27}
\sqrt[3]{64}
\sqrt[3]{-125}
\sqrt[3]{216}
\sqrt[3]{-a^3}
\sqrt[3]{-a^6}
\sqrt[3]{a^9b^3}
\sqrt[3]{343y^3}
\sqrt[3]{512x^{15}}
\sqrt[3]{-729a^3b^9}
\sqrt[3]{1000a^6b^{12}}
\sqrt[3]{-27a^3b^6c^{12}}
\sqrt[3]{8a^{12}b^3c^9}
\sqrt[3]{-125a^6b^6z^6}
\sqrt[3]{x^{3n}}
\sqrt[3]{x^{6n}}
\sqrt[3]{-x^{9n}}
\sqrt[3]{x^{12n}y^{3n}}
\sqrt[5]{x^{15}}
\sqrt[5]{32x^{10}}