## IELTS course, english course, online writingcourses, online english speaking

Search IELTS-Exam.net:

Mathplanet

# 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

$$\fracx5-\sqrtx$$

• 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
•  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

Learning Objectives

 After completing this tutorial, you should be able to: Solve radical equations. Solve equations that have rational exponents.

Introduction

Tutorial

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

 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.

 Example 1: Solve the radical equation . 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.

 *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 .

 *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