Table of Contents

1. Exponents
   1. Basic Exponent Rules
2. Radicals
   1. Simplifying Radicals
   2. Rationalizing Radicals
3. R Examples: Graphing Exponential Functions
4. R Examples: Plotting Radical Functions

Exponents and radicals are fundamental concepts in algebra that relate to the repeated multiplication of a number. Exponents, often called "powers," are a way to express the multiplication of a number by itself a certain number of times. Radicals, commonly known as "roots," are the inverse operation of exponentiation.

Exponents

The expression \( a^n $\)means that the base \( a $\)is multiplied by itself \( n $\)times. For example, \( 3^4 $\)is \( 3 \times 3 \times 3 \times 3 \).

Basic Exponent Rules

• Product Rule: \( a^m \times a^n = a^{m+n} \)
• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power Rule: \( (a^m)^n = a^{m \times n} \)

Radicals

A radical expression like \( \sqrt[n]{a} $\)is the inverse of raising \( a $\)to the power of \( n \). Thus, \( \sqrt[n]{a^n} = a \).

Simplifying Radicals

Simplifying a radical involves finding the root of the number under the radical sign. For example, \( \sqrt{9} = 3 $\)because \( 3^2 = 9 \).

Rationalizing Radicals

Rationalizing a radical means eliminating the radical from the denominator of a fraction. For example, to rationalize \( \frac{1}{\sqrt{2}} \), multiply the numerator and denominator by \( \sqrt{2} $\)to get \( \frac{\sqrt{2}}{2} \).

R Examples: Graphing Exponential Functions

To visualize exponential growth, we can graph an exponential function using R.

library(ggplot2)
x <- seq(-2, 2, by = 0.1)
y <- 2^x
ggplot(data.frame(x, y), aes(x, y)) +
    geom_line() +
    ggtitle("Graph of 2^x") +
    xlab("x") +
    ylab("2^x")

R Examples: Plotting Radical Functions

Similarly, we can graph a simple radical function, like \( y = \sqrt{x} \), to understand its behavior.

library(ggplot2)
x <- seq(0, 10, by = 0.1)
y <- sqrt(x)
ggplot(data.frame(x, y), aes(x, y)) +
    geom_line() +
    ggtitle("Graph of sqrt(x)") +
    xlab("x") +
    ylab("sqrt(x)")

Understanding exponents and radicals is crucial in algebra as they form the basis for more complex topics like logarithms, polynomial equations, and calculus.