Section 2

Powers, exponents, and radicals

fundamentalviz

a^{m} a^{n} = a^{m + n}

When multiplying powers with the same base, the exponents are added.

fundamental

\frac{a ^{m}}{a ^{n}} = a^{m - n} (a \neq = 0)

When dividing powers with the same base, the exponents are subtracted.

fundamental

(a^{m})^{n} = a^{mn}

The exponents are multiplied.

fundamental

(ab)^{n} = a^{n} b^{n}; (a / b)^{n} = a^{n} / b^{n}

A power distributes over products and quotients.

fundamental

a^{0} = 1; a^{- n} = \frac{1}{a ^{n}} (a \neq = 0)

Extends the laws of exponents to zero and negative integers.

fundamental

a^{m / n} = n a^{m}

Relates rational powers and radicals.

In the real numbers, if the root index is even, the radicand must be nonnegative.

fundamental

n ab = n a n b; n a / b = \frac{n a}{n b}

Allows simplifying products and quotients of radicals under the domain conditions.

fundamentalviz

\frac{1}{a + b} = \frac{a - b}{a ^{2} - b}

Removes radicals from the denominator using the conjugate.