Roots and Radicals

Chapter 15

Roots and Radicals

Martin-Gay, Developmental Mathematics 2

15.1 – Introduction to Radicals

15.2 – Simplifying Radicals15.3 – Adding and Subtracting Radicals

15.4 – Multiplying and Dividing Radicals

15.5 – Solving Equations Containing Radicals

15.6 – Radical Equations and Problem Solving

Chapter Sections

§ 15.1

Introduction to Radicals


Square Roots

Opposite 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.


The principal (positive) square root is noted as

a

The negative square root is noted as

a

Principal Square Roots


Radical 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.

Radicands


49 7

1625

45

4 2

Radicands

Example


Square 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 Squares


Radicands 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

Perfect Square Roots

Example


The cube root of a real number a

abba 33 ifonly

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

Cube Roots


3 27 3

3 68x 22x

Cube Roots

Example


Other 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.

nth Roots


Simplify the following.

20225 ba 105ab

39

364b

a3

4ba

nth Roots

Example

§ 15.2

Simplifying Radicals


baab

0b if ba

ba

a bIf and are real numbers,

Product Rule for Radicals


Simplify the following radical expressions.

40 104 102

165

165

45

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


Example



7x xx6 xx6 xx3

16

20x

16

20

x

8

54x 8

52x


Example


nnn baab

0 if nn

nn b

ba

ba

n a n bIf and are real numbers,

Quotient Rule for Radicals



3 16 3 28 33 28 3 2 2

3

643

3

3

643

433


Example

§ 15.3

Adding and Subtracting Radicals


Sums and Differences

Rules 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


In 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.

Like Radicals


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


§ 15.4

Multiplying and Dividing Radicals


nnn abba

0 if b ba

ba

nn

n

n a n bIf and are real numbers,

Multiplying and Dividing Radical Expressions



xy 53 xy15

23

67

ba

ba23

67

baba

44ba 22ba

Multiplying and Dividing Radical Expressions

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.

23

22

3 96

3

3

33

2223

26

33

3

393 6

3

3

273 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.

3223

3323223222323

3232

32

322236

1

322236

322236


Example

§ 15.5

Solving Equations Containing Radicals


Power 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.

Extraneous Solutions


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.

Solving Radical Equations

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.



Solve the following radical 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 4x


Example


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

3 32( ) 1 8 846 true

21 21 142 4 8

8425

221

825

221

8226

falseSo the solution is x = 3.

Example continued



Solve the following radical equation.425 yy

22425 yy

44445 yyy

445 y

445

y

22

445

y

41625

y

1689

16254 y


Example



5 289 891 46 16

16252

16169

452

413

43

413

false So the solution is .

Example continued



Solve the following radical equation.24342 xx

43242 xx 22

43242 xx

43434442 xxx4343842 xxx43412 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


§ 15.6

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). 2

122

12 yyxxd

22 28)2(5 d

22 63 d

5345369 d

The Distance Formula

Example

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Roots and Radicals