### Properties of Radical Functions and Expressions Video 2

How to rewrite radical terms as power terms

This video explains how to simplify power and exponential expressions and solves the following problems to demonstrate the principles. You can try to solve the problems before watching the video and click "Reveal Answer" to see if you are correct.

**Problem:** Simplify the following expressions.

- \(\dfrac{6x^2y^3}{18x^6y^4}\)
- \(\dfrac{2^x\cdot3^4}{2^8\cdot 3^y}\)

Reveal Answer

- \(\dfrac{1}{3x^4y}\)
- \(2^{x-8}\cdot 3^{4-y}\)

How to rewrite radical terms as power terms

Simplifying an expression with both power and exponential functions using properties of exponents

Explaining how to solve equations with power and exponential terms

Properties of exponents applied to power and exponential expressions

Properties of multiplying and simplifying power and exponential expressions

Properties of exponents applied to power and exponential expressions

Simplifying power and exponential expressions using properties of exponents

Simplifying power and exponential expressions using properties of exponents

Properties of exponential growth and decay functions

Properties of the exponential function with base e

Solving equations with exponential terms

Solving equations with base e exponential terms

Solving an equation with exponential functions

Solving an equation with exponential functions

Writing a non-linear function to represent the line graphed on a semilog plot

Using the second derivative to find the maximum rate of change in a word problem

Determining divergence or convergence of improper integrals

Determining divergence or convergence of improper integrals

Finding the domain of a function with a root and a logarithm in the denominator

Explaining how to solve radical equations and then solving example problems

Solving equations with odd radicals

Explaining the terminology and rules of logarithms

Explaining how to solve logarithmic equations

Solving logarithmic equations with a single logarithm term

Solving logarithmic equations with multiple logarithm terms

Solving logarithmic equations with multiple logarithm terms on both sides

Finding the properties of an exponential function including domain, range, end behavior, and intercepts

Finding the properties of an exponential function including domain, range, end behavior, and intercepts

Finding the properties of an exponential function including domain, range, end behavior, and intercepts

Finding the domain of a function that is a fraction containing an exponential and root

Finding the domain of a function that is a fraction containing an exponential and root

Finding the domain of a function that is a fraction containing exponentials and a square root

Calculating how much to invest with continuously compounded interest

Identifying the parent function and transformations for a given graph

Examples of calculating infinite limits with exponentials in a rational function

Examples of finding derivatives and calculating approximate profit

Using integrals to solve application problems

Limits with exponential functions

Determining where functions are continuous

Using the power rule to find derivatives

Examples of differentiating exponential and logarithmic functions

More examples using u-Substitution

Solving more application problems using definite integrals

Application problems finding the average value of a function

Solving examples finding the area between curves

The definition of higher derivatives and solving an example

Using derivatives to find the velocity and acceleration functions from the position function

Explaining the formulas for the derivatives of exponential functions

Calculating the derivative of a function with exponentials using the chain and product rule

Calculating a derivative of a quotient with an exponential function

Explaining exponential functions in discrete time

Solving a population example of exponential growth in discrete time

Solving a population grown example on recurrence equations

Explaining the definition of logarithmic scale and double-log plots

Explaining the logarithmic scale and a semilog plot

Using logarithms to transform a power function into a linear function

Writing a non-linear function for the graph containing two points on a double-log plot

Using logarithms to transform an exponential function to a linear function

Simplifying a fraction using properties of exponents

Simplifying a fraction containing fractional exponents

Simplifying a fraction with negative exponents

Determining properties of an exponential function

Determining properties of an exponential function

Determining properties of an exponential function

Identifying the graph of an exponential function

Solving an equation with an exponential function

Determining properties of an exponential function

Solving an equation with an exponential function

Solving an equation with an exponential function

Solving an equation with an exponential function

Solving an equation with exponential functions by factoring

Solving an equation with logarithmic functions

Solving an equation with logarithmic functions

Solving an equation with logarithmic functions

Solving an equation with logarithmic functions

Solving an equation with logarithmic functions

Modeling a population size with an exponential function

Review of limits and derivatives of inverse trigonometric functions

Derivatives of exponential functions and the exponential model

Review of limits and derivatives of inverse trigonometric functions

Mean Value Theorem and using derivatives to find the shape of curves

Antiderivatives and physics applications

Approximation and Newton's Method, and limits and derivatives of exponential functions

Approximation and Newton's Method, and limits and derivatives of exponential functions

Derivatives of exponential and logarithmic functions and the exponential model

Derivatives of exponential and logarithmic functions and the exponential model

Mean Value Theorem and properties of a graph

Mean Value Theorem and properties of a graph

Using L'Hospital's Rule to solve limits

Using Reimann sums and the Fundamental Theorem of Calculus

Using Reimann sums and the Fundamental Theorem of Calculus

Examples of integration by substitution

Differentiating a function with an exponential function and using the Power Rule

Using the Product Rule to differentiate a function

Using the Quotient Rule to differentiate a function

Using the Product and Quotient Rules to differentiate a function

Using the Product Rule to find a first and second derivative

Using the Chain Rule to differentiate functions with exponential and trigonometric functions

Using the Chain Rule to find a first and second derivative

Using the Chain Rule to find the pattern for a higher order derivative

Finding the derivative using implicit differentiation

Using logarithmic differentiation to find the derivative of a function

Using logarithmic differentiation to find the derivative of a function

Using logarithmic differentiation to find the derivative of a function

Using the Product and Chain Rule to find several derivatives

Using the Product, Quotient and Chain Rules to find several derivatives

Using the Quotient and Chain Rules to find several derivatives

Using the Product and Chain Rules to differentiate functions with logarithms

Differentiating functions with logarithmic and inverse trigonometric functions

Differentiating functions with inverse trigonometric functions

Finding the derivative of an implicit function

Finding the derivative of an implicit function

Evaluating a limit using L'Hospital's Rule

Evaluating a limit with an indeterminate product using L'Hospital's Rule

Evaluating a limit with an indeterminant difference using L'Hospital's Rule

Evaluating a limit with an indeterminate power using L'Hospital's Rule

Evaluating a limit with an indeterminant power using L'Hospital's Rule

Antidifferentiating twice to find a function from its second derivative

Antidifferentiating twice to find a function from its second derivative

Antidifferentiating to find a function from its derivative

Antidifferentiating twice to find a function from its second derivative

Antidifferentiating twice to find a function from its second derivative

Using antidifferentiation to find the height of a cliff from the impact speed of a dropped stone

Evaluating a limit using L'Hospital's Rule

Evaluating a limit with an indeterminant power using L'Hospital's Rule

Evaluating a limit with an indeterminant product using L'Hospital's Rule

Finding the antiderivative of a function and using a function value to find the constant

Evaluating the limit at infinity of a fraction with exponential functions

Evaluating the limit at negative infinity of a fraction with exponential functions

Finding a derivative using the Chain Rule and Product Rule

Using L'Hospital's Rule to solve a limit with an indeterminant power

Evaluating a definite integral with an exponential and sine function

Examples reviewing integration techniques from Calculus I

Review of improper integrals and the Comparison Theorem

Review of Taylor and Maclaurin Series and their properties

Review of Taylor Polynomials and Taylor's Inequality

Finding Taylor and Maclaurin Series for functions

Examples of integration by substitution

Using the disk, washer, and shell method to find a volume of revolution

Finding volume using using disks, washers and the shell method

Examples using integration by parts

Integrating an exponential function using u-substitution

Finding the area of a region bounded by curves

Evaluating an integral using integration by parts

Evaluating an integral using integration by parts

Using the comparison test to determine if an improper integral converges or diverges

Determining if a series converges or diverges and find the sum if it converges

Using the comparison test to determine if a series converges or diverges

Using the limit comparison test to determine if a series converges or diverges

Determining if a series is absolutely convergent, conditionally convergent, or divergent.

Finding the Taylor polynomial of a function

Finding the length of a curve given by parametric equations

Finding the area of a region defined using polar coordinates

Solving an indefinite integral with an exponential function using integration by parts

Solving a definite integral with an exponential function using integration by parts

Solving an indefinite integral with an exponential function using integration by parts

Using integration by parts twice for an integral with sine and an exponential function

Using u-substitution on an indefinite integral with an exponential function, sine, and cosine

Using u-substitution to solve an indefinite integral with an exponential function

Review of limits and derivatives of inverse trigonometric functions

Derivatives of exponential functions and the exponential model

Review of limits and derivatives of inverse trigonometric functions

Mean Value Theorem and using derivatives to find the shape of curves

Antiderivatives and physics applications

Approximation and Newton's Method, and limits and derivatives of exponential functions

Using the limit definition to find a derivative and proving the power rule

Using the limit definition to find a derivative and proving the power rule

Examples involving the tangent line to an exponential function and finding the derivative of hyperbolic cosine

Approximation and Newton's Method, and limits and derivatives of exponential functions

Using implicit differentiation to solve a problem and to prove the power rule for rational exponents

Proving facts about logarithms and exponentials including the derivative of an exponential with an arbitrary base

Proving the derivative of ln(x) and the power and product rules using logarithmic differentiation

Derivatives of exponential and logarithmic functions and the exponential model

Mean Value Theorem and properties of a graph

Mean Value Theorem and properties of a graph

Using L'Hospital's Rule to solve limits

Using Reimann sums and the Fundamental Theorem of Calculus

Using Reimann sums and the Fundamental Theorem of Calculus

Examples of integration by substitution

Examples reviewing integration techniques from Calculus I

Review of improper integrals and the Comparison Theorem

Review of Taylor and Maclaurin Series and their properties

Review of Taylor Polynomials and Taylor's Inequality

Finding Taylor and Maclaurin Series for functions

Examples of integration by substitution

Using the disk, washer, and shell method to find a volume of revolution

Finding volume using using disks, washers and the shell method

Examples using integration by parts

Finding the limit of a three-dimensional vector function

Finding the partial derivatives of a function of two variables

Finding all second-order partial derivatives of a function of two variables

Evaluating a double integral over a given region

Evaluating a double integral by reversing the order of integration

Evaluating a line integral along a curve

Evaluating a line integral along a curve in space

Finding the curl and divergence of a three dimensional vector field

Evaluating a surface integral of a vector field

Using Python to approximate a definite integral using left endpoint Riemann sums

Performing basic numerical calculations in Python

Showing how the exponential function is the solution to an initial value problem

Finding the domain of a function with a denominator, square root, and logarithm

Finding the domain of a function with a natural logarithm and denominator

Finding the domain of a function with a square root in the denominator

Finding the domain of a logarithmic function

Finding the domain of even and odd roots

Finding the domain of functions with even and odd roots

Finding the domain of several rational functions

Finding the domain of a rational function

Explaining how to find any holes and vertical asymptotes of a rational function

Finding the domain of a rational function using factoring

Explaining domain restrictions for denominators, even roots, and logarithms

Explaining the main three forms for linear equations

Discussing properties of linear functions

Discussing the types of linear graphs including horizontal and vertical lines and their properties

Discussing how to write the equation of a line through two given points

Writing a linear equation of profit from a business word problem

Solving a linear equation algebraically and showing how this relates to the graph

Solving a linear equation algebraically

Solving a linear equation algebraically and showing how this relates to the graph

Explaining the format and possible solutions for a system of two linear equations

Solving a system of linear equations with two variables

Solving a system of two linear equations with two variables using the elimination method

Solving a system of two linear equations using a TI-84 calculator

Solving a system of two linear equations using a TI-84 calculator

Solving a system of two linear equations using a TI-84 calculator

Explaining the standard form for a quadratic equation and the possible number of solutions

Solving quadratic equations with the difference of two squares formula

Solving quadratic equations by factoring

Solving a quadratic equation by factoring

Solving a quadratic equation by factoring

Solving quadratic equations with the quadratic formula and discussing the number of possible solutions

Explaining when a quadratic equation does not have a real solution

Explaining the graph and properties for the parent function of quadratics

Discussing properties of quadratic functions including the vertex, domain, range, and end behavior

Solving equations with multiple radicals

Solving equations with even roots and checking the solutions

Explaining the general properties of even and odd root functions

Explaining the basic properties of logarithmic functions

Expanding logarithmic expressions using logarithm rules

Condensing a logarithmic expression into a single logarithm

Discussing parent functions of rational functions and their properties

Finding horizontal asymptotes for rational functions

Explaining how to do polynomial long division

Finding horizontal asymptotes for rational functions

Determining the end behavior and horizontal asymptotes for rational functions

Explaining basic algebraic operations for fractions

How to simplify rational expressions

Adding and subtracting rational expressions by finding a common denominator

Explaining how to multiply and divide rational expressions

Performing algebraic operations with rational expressions and simplifying the answer

Solving equations that contain rational expressions

Solving an equation that contains rational expressions

Solving an equation that contains rational expressions

Explaining the format and possible solutions for a system of two linear equations