Roots and Radicals

Roots and RadicalsIntroduction to Radicals

1 – Introduction to Radicals

2 – Simplifying Radicals

3 – Adding and Subtracting Radicals

4 – Multiplying and Dividing Radicals

5 – Solving Equations Containing Radicals

6 – Radical Equations and Problem Solving

Sections

Square RootsOpposite of squaring a number is taking the square root of a number.

A number b is a square root of a number a if b2 = a.

In order to find a square root of a, you need a # that, when squared, equals a.

Principal Square Roots

The principal (positive) square root is noted as

a

The negative square root is noted as

a

RadicandsRadical expression is an expression containing a radical sign.

Radicand is the expression under a radical sign.

Note that if the radicand of a square root is a negative number, the radical is NOT a real number.

49 7

16

25

4

5

4 2

RadicandsExampl

e

Perfect SquaresSquare roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers).

Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrational numbers.

IF REQUESTED, you can find a decimal approximation for these irrational numbers.

Otherwise, leave them in radical form.

Perfect Square RootsRadicands might also contain variables and powers of variables.To avoid negative radicands, assume for this chapter that if a variable appears in the radicand, it represents positive numbers only.

1064x 58x

Example

Cube RootsThe cube root of a real number a

abba 33 ifonly

Note: a is not restricted to non-negative numbers for cubes.

3 27 3

3 68x 22x

Cube RootsExampl

e

nth RootsOther roots can be found, as well.The nth root of a is defined as

abba nn ifonly

If the index, n, is even, the root is NOT a real number when a is negative.

If the index is odd, the root will be a real number.

Simplify the following.

20225 ba 105ab

39

364

b

a3

4

b

a

Example

Sim

plifyi

ng

Rad

ical

s

baab

0b if b

a

b

a

a bIf and are real numbers,

Product Rule for Radicals

Simplify the following radical expressions.

40 104 102

16

5 16

5

4

5

15No perfect square factor, so the radical is already simplified.

Simplifying RadicalsExampl

e


7x xx6 xx6 xx3

16

20

x

16

20

x

8

54

x 8

52

x


e

nnn baab

0 if n

n

n

n bb

a

b

a

n a n bIf and are real numbers,

Quotient Rule for Radicals


3 16 3 28 33 28 3 2 2

3

64

3 3

3

64

3

4

33


e

Add

ing

and

Subt

ract

ing

Rad

ical

s

Sums and DifferencesRules in the previous section allowed us to split radicals that had a radicand which was a product or a quotient.We can NOT split sums or differences.

baba

baba

Like RadicalsIn previous chapters, we’ve discussed the concept of “like” terms.

These are terms with the same variables raised to the same powers.

They can be combined through addition and subtraction.

Similarly, we can work with the concept of “like” radicals to combine radicals with the same radicand.Like radicals are radicals with the same index and the same radicand.Like radicals can also be combined with addition or subtraction by using the distributive property.

373 38

24210 26

3 2 42 Can not simplify

35 Can not simplify

Adding and Subtracting Radical Expressions

Example

Simplify the following radical expression.

331275

3334325

3334325

333235

3325 36

Example

Simplify the following radical expression.

91464 33

9144 3 3 145

Example

Simplify the following radical expression. Assume that variables represent positive real numbers.

xxx 5453 3 xxxx 5593 2

xxxx 5593 2

xxxx 5533

xxxx 559

xxx 59 xx 510

Example

Multiplying and Dividing Radicals

nnn abba

0 if b b

a

b

an

n

n

n a n bIf and are real numbers,

Multiplying and Dividing Radical Expressions


xy 53 xy15

23

67

ba

ba

23

67

ba

ba44ba 22ba

Example

Many times it is helpful to rewrite a radical quotient with the radical confined to ONLY the numerator.

If we rewrite the expression so that there is no radical in the denominator, it is called rationalizing the denominator.

This process involves multiplying the quotient by a form of 1 that will eliminate the radical in the denominator.

Rationalizing the Denominator

Rationalize the denominator.

2

3

2

2

3 9

6

3

3

3

3

22

23

2

6

33

3

39

3 6

3

3

27

3 6

3

3 6 33 3 2

Example

Many rational quotients have a sum or difference of terms in a denominator, rather than a single radical.

In that case, we need to multiply by the conjugate of the numerator or denominator (which ever one we are rationalizing).

The conjugate uses the same terms, but the opposite operation (+ or ).

Conjugates

Rationalize the denominator.

32

23

332322

3222323

32

32

32

322236

1

322236

322236

Rationalizing the Denominator

Example

Solving

Equations

Containing Radicals

Extraneous SolutionsPower Rule (text only talks about squaring, but applies to other powers, as well).

If both sides of an equation are raised to the same power, solutions of the new equation contain all the solutions of the original equation, but might also contain additional solutions.

A proposed solution of the new equation that is NOT a solution of the original equation is an extraneous solution.

Solving Radical Equations

Solve the following radical equation.

51 x

2251 x

251x

24x

24 1 5

525 true

Substitute into the original equation.

So the solution is x = 24.

Example


55 x

2255 x

255 x

5x

5 55 525

Does NOT check, since the left side of the equation is asking for the principal square root.So the solution is .


Example

Steps for Solving Radical Equations1) Isolate one radical on one side of equal

sign.2) Raise each side of the equation to a power

equal to the index of the isolated radical, and simplify. (With square roots, the index is 2, so square both sides.)

3) If equation still contains a radical, repeat steps 1 and 2. If not, solve equation.

4) Check proposed solutions in the original equation.


011 x

11 x

2211 x

11x

2x

2 1 1 0 011

011 true


So the solution is x = 2.

Example


812 xx

xx 281

22281 xx

2432641 xxx 2433630 xx )421)(3(0 xx

213 or 4

x

Example

Substitute the value for x into the original equation, to check the solution.

3 32( ) 1 8 846 true

21 21 14

24

8

84

25

2

21

82

5

2

21

82

26 falseSo the solution is x =

3.

Example continued


Solve the following radical equation.425 yy

22425 yy

44445 yyy

445 y

44

5 y

22

44

5

y

416

25y

16

89

16

254 y

Example


5 289 891

46 16

16

252

16

169

4

52

4

13

4

3

4

13 false So the solution is

.

Example continued



24342 xx

43242 xx

2243242 xx

43434442 xxx

4343842 xxx

43412 xx

22 43412 xx6448)43(16144242 xxxx

080242 xx

0420 xx

20or 4x

Example


2( ) 4 3( 4 24 4) 2164

242

true

2( ) 4 3( ) 420 20 2

26436

286

true

So the solution is x = 4 or 20.

Example continued


Radical Equations

and Problem Solving

Pythagorean TheoremIn a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.

(leg a)2 + (leg b)2 = (hypotenuse)2

leg ahypotenuse

leg b

The Pythagorean Theorem

Find the length of the hypotenuse of a right triangle when the length of the two legs are 2 inches and 7 inches.

c2 = 22 + 72 = 4 + 49 = 53

53c = inches

Using the Pythagorean Theorem

Example

By using the Pythagorean Theorem, we can derive a formula for finding the distance between two points with coordinates (x1,y1) and (x2,y2).

212

212 yyxxd

The Distance Formula

Find the distance between (5, 8) and (2, 2).

212

212 yyxxd

22 28)2(5 d

22 63 d

5345369 d

The Distance FormulaExampl

e

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Roots and Radicals