1 Exponents and Radicals Chapter 11. 2 Flashback: One semester ago……..

1

Exponents and Radicals

Chapter 11

2

RULES FOR INTEGRAL EXPONENTS: m, n positive real numbers

m nx x

0m

n

xx

x

( )m nx

( )nxy

0n

xy

y

0 0x x

0nx x

Flashback: One semester ago……..

3

Simplifying Expressions with Integral ExponentsHelpful Tip: Watch the order of operations ….or else!

Examples

22

00

11

5 5

3 3

8 8

a a

m m

4

Sect 11.1 : Simplifying Expressions with Integral ExponentsUse one or a combination of the laws of exponents to simplify each expression. State final answers using only positive exponents. Assume all variables represent positive values.

3 02 4 5 1 3

1) 3 2)2 4

x y x a b

5

Simplifying Expressions with Integral Exponents

32 5 23) 4 4)m x y

6


3

5

3 45) 6)

5b

7


3 1 0

327)

2

y y y

y

8


23

2 2

28)

3

ab

a b

9


27

315

9) 10) 3

mz

n

10


1 2211) 12) 3 2m n a b

11


1 1

2 213)

x y

x y

12


1 314) 3 2 x x y

13

Sect 11.2: Fractional (Rational) Exponents

RATIO NAL EXPO NENTS:

nn xx 1

n mmnn

m

xxx

438

power

root

Note: These properties are valid as long as does not involve the even root of a negative number.

n a

14

Fractional (Rational) Exponents

When evaluating expressions involving fractional exponents without a calculator:

It is usually best to find the root first, as indicated by the denominator, then raise it to the power indicated in the numerator.

2 33 427 16

15

Fractional (Rational) Exponents

You can evaluate expressions involving fractional exponents using the calculator. Use the key and type parentheses around the fraction.

Example: Evaluate using the TI-84. 341296

16

Simplifying Expressions Involving Fractional Exponents

The same laws of exponents apply to fractional exponents, though the work may be a little messier.

23

12

2 1

3 21)

2)

x x

x

x

17


4

2 33

12 23

12

3)

4) 64

x

x y

18


14

34

13

1

65)

2 6

6) 125

19


2 13 4

25

310

2

7) 1000 81

8)R R

R

20

Sect 11.3: Simplifying Radicals

Let a and b represent positive real numbers.

nn n n

n n n

m n mn

n

nn

a a a

a b ab

a a

a a

bb

21

Simplifying Radicals

Remember ….

is NOT equivalent to writingn n na b a b

9 16 9 16

22


To reduce a radical to simplest form:

1. Remove all perfect nth power factors from a radical of order n.

2. If a fraction appears under the radical or there is a radical in the denominator of the expression, simplify by rationalizing the denominator.

3. If possible, reduce the order of the radical.

23


Removing all nth power factors

3 18

3 5 12

1) 12

2) 54

x y

a b c

24


Reducing the order of the radical

9

6 2 16

3) 27

4) 25a b

25


Rationalizing the denominator

3

55)

2

26)

3

a

b

26


Mixed bag…

6 7

2 1

7) 98

8)

x y

a b

27

Sect 11.4: Addition and Subtraction of Radicals

To perform addition or subtraction of radicals, you combine like (similar) radicals.

Similar radicals differ only in their numerical coefficients (same radicand AND same index).

32 3 3 2 5 3 2

28

Addition and Subtraction of Radicals

Plan of Action:

1. Express each radical in simplest form.

2. Combine like radicals.

1) 3 75 2 12 2 48 Examples

29


3 3 42) 32 50 18x x x

30


3 53 33) 24 3 4) 9 50 4 72x x x x x

31


25) 6 18

3

32

Sect 11.5: Multiplication & Division of Radicals

To multiply expressions containing radicals, we will use the property

where a and b represent positive values.

Notice that the order (indexes) of the radicals being multiplied must be the same.

n n na b ab

33

Multiplication of Radicals

Check it out!

1) 5 3 4 6

2) 4 2 3 5 2 8

34

Multiplication of Radicals

3) 5 2 3 5 3

4) 7 3 2 7 3 2

35

Division of Radicals

5

3 2Rationalize the denominator:

We saw earlier that when dealing with an expression containing a

radical in the denominator, we had to rationalize the denominator to

write it in simplest form.

We will now deal with denominators containing two terms.

36

Division of Radicals

3 5

2 15 5

15Rationalize the denominator:

To rationalize a denominator that is the sum or difference of two terms, multiply the numerator and denominator of the fraction by the _____________________ of the denominator.

37

Sect 14.4 Solving Radical Equations

2

33

nn

x x

x x

x x

To solve radical equations, we will use the fact that

That is, we will apply the appropriate inverse operation to “get the variable out of the radical”.

38

Solving Radical Equations

To solve a radical equation involving one radical:

1. Isolate the radical expression on one side of the equation.

2. Raise both sides of the equation to the power that is the same as the order of the radical (inverse operation).

3. Solve the resulting equation for the variable.

4. Check for extraneous solutions* by checking the apparent solutions in the original equation.

*Extraneous solutions may be introduced when both sides of an equation are raised to an even power.

39


2 2 3 10 16x

Example 1

Solve:

40


4 1 8 9x x

Example 2

Solve:

41


3- 7 7 -1 40 19n

Example 3

Solve:

42

WIND POWER The power generated by a windmill is related to the velocity of the wind by the formula

where P is the power (in watts) and v is the velocity of the wind (in mph). Find how much power the windmill is generating when the wind is 29 mph.


3

0.02

Pv

43

To solve a radical equation involving two square roots:

1. Isolate one of the radical expressions on one side of the equation.

2. Square both sides of the equation.

3. Simplify.

4. Isolate the remaining radical expression on one side of the equation.

5. Square both sides of the equation.

6. Solve the resulting equation for the variable.

7. Check for extraneous solutions by checking the apparent solutions in the original equation.

44

Solving Radical Equations Involving Two Radicals

Example 1

Solve: 4d+1 6 1d

45

Solving Radical Equations Involving Two Radicals

Example 2

Solve: 2 +5 2 1 4 0x x

46

End of Section

1 Exponents and Radicals Chapter 11. 2 Flashback: One semester ago……..

Documents

Transcript of 1 Exponents and Radicals Chapter 11. 2 Flashback: One semester ago……..