st philip’s christian college fees

ielts-exam.net

IELTS course, english course, online writing
courses, online english speaking

Search IELTS-Exam.net:

Solving Radical Equations Simplifying Radical Expressions





Mathplanet


Menu

Simplify radical expressions

The properties of exponents, which we’ve talked about earlier, tell us among other things that

$$\beginpmatrix xy \endpmatrix^a=x^ay^a$$

$$\beginpmatrix \fracxy \endpmatrix^a=\fracx^ay^a $$

We also know that

$$\sqrt[a]x=x^\frac1a$$

$$or$$

$$\sqrtx=x^\frac12$$

If we combine these two things then we get the product property of radicals and the quotient property of radicals. These two properties tell us that the square root of a product equals the product of the square roots of the factors.

$$\sqrtxy=\sqrtx\cdot \sqrty$$

$$\sqrt\fracxy=\frac\sqrtx\sqrty$$

$$where\:\: x\geq 0,y\geq 0 $$

The answer can’t be negative and x and y can’t be negative since we then wouldn’t get a real answer. In the same way we know that

$$\sqrtx^2=x\: \: where\: \: x\geq 0$$

These properties can be used to simplify radical expressions. A radical expression is said to be in its simplest form if there are

no perfect square factors other than 1 in the radicand

$$\sqrt16x=\sqrt16\cdot \sqrtx=\sqrt4^2\cdot \sqrtx=4\sqrtx$$

no fractions in the radicand and

$$\sqrt\frac2516x^2=\frac\sqrt25\sqrt16\cdot \sqrtx^2=\frac54x$$

no radicals appear in the denominator of a fraction.

$$\sqrt\frac1516=\frac\sqrt15\sqrt16=\frac\sqrt154$$

If the denominator is not a perfect square you can rationalize the denominator by multiplying the expression by an appropriate form of 1 e.g.

$$\sqrt\fracxy=\frac\sqrtx\sqrty\cdot \colorgreen \frac\sqrty\sqrty=\frac\sqrtxy\sqrty^2=\frac\sqrtxyy$$

Binomials like

$$x\sqrty+z\sqrtw\: \: and\: \: x\sqrty-z\sqrtw$$

are called conjugates to each other. The product of two conjugates is always a rational number which means that you can use conjugates to rationalize the denominator e.g.

$$\fracx4+\sqrtx=\fracx\left ( \colorgreen 4-\sqrtx \right )\left ( 4+\sqrtx \right )\left ( \colorgreen 4-\sqrtx \right )= $$

$$=\fracx\left ( 4-\sqrtx \right )16-\left ( \sqrtx \right )^2=\frac4x-x\sqrtx16-x$$


Video lesson

Simplify the radical expression

$$\fracx5-\sqrtx$$


Share on Facebook

  • Rational expressions


    • Algebra 1

    • Rational expressions

      • Overview
      • Simplify rational expression
      • Multiply rational expressions
      • Division of polynomials
      • Add and subtract rational expressions
      • Solving rational expressions
  • Algebra 2


    All courses

      • Algebra 2
        Overview
  • Geometry


    All courses

      • Geometry
        Overview
  • SAT


    All courses

      • SAT
        Overview
  • ACT


    All courses

      • ACT
        Overview
  • West Texas A&M University - Home
    Virtual Math Lab
    • VML
    • COLLEGE ALGEBRA
    • INTERMEDIATE ALGEBRA
    • BEGINNING ALGEBRA
    • GRE MATH
    • THEA/ACCUPLACER
    College Algebra
    Tutorial 19: Radical Equations and
    Equations Involving Rational Exponents


    WTAMU > Virtual Math Lab > College Algebra

     

    deskLearning Objectives





    After completing this tutorial, you should be able to:

    1. Solve radical equations.
    2. Solve equations that have rational exponents.

    desk Introduction



     

    In this tutorial, we will be looking at solving two different types
    of equations, radical equations and equations that have rational exponents. 
    Both of these equations have the same ultimate goal, to get your variable
    on one side and everything else on the other side using inverse operations.  
    Also, after removing the radical or rational exponent in the equations
    in this tutorial, they become either a linear or quadratic equation. 
    Good news and bad news,  as mentioned in other tutorials, a lot of
    times in math you use previous knowledge to help work the new concepts. 
    That is good because you do not have to approach the problem as totally
    new and learn all new steps.  That can be overwhelming.  It is
    bad because you do have to remember things from the past.  Sometimes
    we condition ourselves to drain our brains after taking a test and sometimes
    forget what we have learned.  If you need a review on radicals in
    general, feel free to go to Tutorial
    4: Radicals
    .  If you need a review on rational exponents in
    general, feel free to go to Tutorial
    5: Rational Exponents
    .  If you need a review on solving linear
    equations, feel free to go to Tutorial
    14: Linear Equations in One Variable
    .  If you need a review
    on solving quadratic equations, feel free to go to Tutorial
    17:  Quadratic Equations
    .  After going through this page,
    you should be an old pro at working with roots.  I think you are ready
    to tackle these equations.

     

     

    desk Tutorial



     


    Solving Radical Equations



     

    Step 1:  Isolate
    one of the radicals. 

     

    In other words, get one radical on one side and everything else on the
    other using inverse operations.

    In some problems there is only one radical.  However, there are
    some problems that have more than one radical.  In these problems
    make sure you isolate just one.

     

    Step 2: Get rid
    of your radical sign. 

     

    The inverse operation to a radical or a root is to raise it to an exponent. 
    Which exponent?  Good question, it would be the exponent that matches
    the index or root number on your radical. 

    In other words, if you had a square root, you would have to square it
    to get rid of it.  If you had a cube root, you would have to cube
    it to get rid of it,  and so forth. 

    You can raise both sides to the 2nd power, 10th power, hundredth power,
    etc.  As long as you do the same thing to both sides of the equation,
    the two sides will remain equal to each other. 

     

    Step 3: If you
    still have a radical sign left, repeat steps 1 and 2.
     
     

    Sometimes you start out with two or more radicals in your equation. 
    If that is the case and you have at least one nonradical term, you will
    probably have to repeat steps 1 and 2.

     

    Step 4: Solve the
    remaining equation. 

     

    The equations in this tutorial will lead to either a linear or a quadratic
    equation.

    If you need a review on solving linear equations, feel free to go to Tutorial
    14: Linear Equations in One Variable

    If you need a review on solving quadratic equations, feel free to go
    to Tutorial 17:  Quadratic Equations .

     

    Step 5:  Check
    for extraneous solutions.

     

    When solving radical equations, extra solutions may come up when you
    raise both sides to an even power.  These extra solutions are called
    extraneous solutions. If a value is an extraneous
    solution, it is not a solution to the original problem.

    In radical equations, you check for extraneous solutions by plugging
    in the values you found back into the original problem. If the left side
    does not equal the right side, then you have an extraneous solution. 



     
     

    notebookExample
    1
    : Solve the radical equation example 1a.

    video View a video of this example



     

    Step 1:  Isolate
    one of the radicals. 



     

    The radical in this equation is already isolated.



     

    Step 2: Get
    rid of your radical sign.



     

    If you square a square root, it will disappear.  This is what
    we want to do here so that we can get x out
    from under the square root and continue to solve for it.



     

    example 1b
    *Inverse of taking the sq. root is squaring
    it



     

    Step 3: If you
    still have a radical left, repeat steps 1 and 2.



     

    No more radicals exist, so we do not have to repeat steps 1 and 2.



     

    Step 4: Solve
    the remaining equation. 



     

    In this example, the equation that resulted from squaring both sides
    turned out to be a linear equation.

    If you need a review on solving linear equations, feel free to go to Tutorial
    14: Linear Equations in One Variable



     

    example 1c
    *Inverse of add. 5 is sub. 5
     

    *Inverse Continue reading “Solving Radical Equations Simplifying Radical Expressions”

    Beith, Spiers School As an old spierian spiers school in beith

    • Flights

    • Vacation Rentals

    • Restaurants

    • Things to do

      Conditions, Triggers, and Event pages Events activated by another event? :: RPG Maker VX Ace How To …









      О сервисе
      Прессе
      Правообладателям
      Связаться с нами
      Авторам
      Рекламодателям
      Разработчикам

      Условия использования
      Конфиденциальность
      Правила и безопасность
      Новые функции

      © 2018 YouTube, LLC