1.3 Radicals and rational exponents  (Page 4/11)

Understanding n Th roots

Suppose we know that $a^3=8.$ We want to find what number raised to the 3rd power is equal to 8. Since $2^3=8,$ we say that 2 is the cube root of 8.

The n th root of $a$ is a number that, when raised to the n th power, gives $a.$ For example, $\mathrm{-3}$ is the 5th root of $\mathrm{-243}$ because ${(\mathrm{-3})}^5=\mathrm{-243.}$ If $a$ is a real number with at least one n th root, then the principal n th root    of $a$ is the number with the same sign as $a$ that, when raised to the n th power, equals $a.$

The principal n th root of $a$ is written as $\sqrt[n]{a},$ where $n$ is a positive integer greater than or equal to 2. In the radical expression, $n$ is called the index    of the radical.

Using rational exponents

Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index $n$ is even, then $a$ cannot be negative.

We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an n th root. The numerator tells us the power and the denominator tells us the root.

All of the properties of exponents that we learned for integer exponents also hold for rational exponents.