Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Roots and Radicals: Solving Power Equations Algebraically

Solving a Power Equation

The objective of solving a power equation such as: 

1.0725 · x0.808 =10,

is to find the numerical value of x that works in this equation.

Solving a Power Equation Using Algebra

You can also solve power equations using algebra, and although the manipulations for solving power equations are not as straightforward as the manipulations for solving linear equations, using algebra is often easier than using the calculator.

In order to use algebra to solve a power equation, you need to be able to use one of the Laws of Exponents:

(x p )q = x p·q (Multiply the exponents.)


Locating the Intersection of Power Functions

When you were working with linear functions, one of the things that you learned how to calculate was the x- and y-coordinates of the point (the intersection point) where the graphs of two linear functions met.

Power functions also intersect, and you can find the x- and y-coordinates of the intersection point using both a graphing calculator and algebra. In order to use algebra to calculate the x-coordinate of the point where two power functions meet, you have to use two of the Laws of Exponents: 

(x p )q = x p·q (Multiply the exponents.) 

(Subtract exponents.)



Locating the Intersection Point of Power Functions Using Algebra

Find the x- and y-coordinates of the intersection point of the two power functions: 

y =100· x-2 and  y = 6.25 · x2.


100 · x-2 = 6.25 · x2 (Set the two equations equal.) 

(Divide both sides by 6.25.) 

16 · x-2 = x2 (Simplify both sides.) 

(Divide both sides by x-2.) 

16 · x-2-(-2) = x2-(-2) (Simplify the fractions with Law of Exponents.) 

16 = x4 (Simplify both sides.) 

(Take the  14 power of both sides.) 

(Simplify with the Law of Exponents.) 

x = 2 (Work out the rest on a calculator.)

To work out the y-coordinate, you can plug x = 2 into either of the power equations given in the problem. 

y =100 · (2)-2 = 25.

Radical Notation

An alternative notation, called radical notation, is sometimes used when writing algebraic expressions and formulas that include fractional powers.

For example, a fractional power such as: ,

is sometimes written as:  .

A fractional power such as:  ,

would be written in radical notation as:  .

If you are ever given an algebraic expression or formula expressed using radical notation, it is best to convert the radicals to fractional powers and then simplify them as much as possible using the laws of exponents. If you have to express your answer in radical notation, then you can convert the answer back to radical notation when you have completed all of your simplifications and manipulations.



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