1.2 – Exponents and Radicals

Video 1 of 4: Zero and Negative Exponents

Video 2 of 4: Laws of Exponents

Video 3 of 4: Scientific Notation

Video 4 of 4: Radicals and Root Properties

Exponents and Radicals

Exponential Notation

Exponential notation is the mathematical way of writing a repeated multiplication. For example, $3\cdot3\cdot3\cdot3$ can be written in exponential notation as $3^4$ indicating that there are four factors of 3. In general, exponential notation is defined as follows:

$a^n=\underbrace{a\cdot a\cdot a \hdots a}_{n \text{ factors}}$

The number $a$ is called the base. The number $n$ is called the exponent.

Zero and Negative Exponents

An exponent need not be a positive number. In fact, exponents can also be zero, or negative. We define these cases by:
If $x\not=0$ is any real number, and $a$ is any positive integer, then

$x^0=1 \ \ \ \ \text{ and } \ \ \ \ x^{-a}=\frac{1}{x^a}$

Laws of Exponents

The nature of exponents lends to the following rules, or laws, of exponents.

$\begin{array}[c]{|l|p{8.2cm}|}\hline 1. x^ax^b=x^{a+b} & To multiply two numbers with the same base, add the exponents. \\ \hline 2. \frac{x^a}{x^b}=a^{a-b} & To divide two numbers with the same base, subtract the exponents. \\ \hline 3. (x^a)^b=x^{ab} & Raising a number with a power to another power multiplies the powers. \\ \hline 4. (xy)^a=x^ay^a & A product raised to a power is the same as the product of each number raised to the power. \\ \hline 5. \left(\frac{x}{y}\right)^a=\frac{x^a}{y^a} & The quotient raised to a power is the quotient of each number raised to that power. \\ \hline 6. \left(\frac{x}{y}\right)^{-a}=\left(\frac{y}{x}\right)^a & Raising a fraction to a negative power is the same as raising the inverse to the positive power. \\ \hline 7. \frac{x^{-a}}{y^{-b}}=\frac{y^b}{x^a} & A factor in a quotient with a negative exponent can interchange its place and have its exponent's signed changed. \\ \hline\end{array}$

Scientific Notation

Exponents are used in the expression of very large or very small numbers. For example, the number 125,000,000,000 can be written $1.25\times 10^{11}. A number in Scientific Notation is expressed [latex]a\times 10^n$