Worksheet 31 (6.1)

Chapter 6 Exponents and Radicals

6.1 Using Integers as Exponents

Summary 1:

Warm-up 1. Simplify:

a) 250 = c) (x2y)0 =

b) (-92)0 = d) -250 =

Problems - Simplify:

1. 80 2. (-10)0

3. -100 4. (ab2)0

Warm-up 2. Simplify:a)

b)

c)

If b is a nonzero real number, then:

If n is a positive integer and b is a nonzero real number, then:

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Problems - Simplify:5. 6. 7.

Worksheet 31 (6.1)

Summary 2:

We can now restate the properties of exponents, presented in Chapter 3, to include all integer exponents. We will also now name the properties.

If m and n are integers and a and b are real numbers, except b 0, whenever it appears in a denominator, then:

1. product of two powers2. power of a power3. power of a product

4. power of a quotient

5. quotient of two powers

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Warm-up 3. Simplify:a) b) c)

d)

e)

Problems - Simplify:

8. 9. 10.

11. 12. Worksheet 31 (6.1)

Warm-up 4. Perform the indicated operations and/or simplify. Express your results using positive integral exponents only.

a)

b)

c)

d)

e)

= =

=

Problems - Perform the indicated operations and/or simplify. Express your results using positive integral exponents only.

122

13. 14.

15. 16.

17. Worksheet

31 (6.1)Warm-up 5. a) Simplify:

=

=

=

b) Express as a single fraction involving positive exponents only:

=

=

Problems18. Simplify:

19. Express as a single fraction involving positive exponents only:

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Worksheet 32 (6.2)

6.2 Roots and Radicals

Summary 1:

Warm-up 1. Evaluate:a) = c) = b) = d) =

Problems- Evaluate:

To square a number means to use the number as a factor twice.

EX. 52 = 55 = 25 and (-7)2 = (-7)(-7) = 49

The square root of a number is one of its two equal factors.EX. 5 is a square root of 25 because 55 = 25,

-5 is a square root of 25 because (-5)(-5) = 25.

The symbol is called a radical sign and is used to designate the nonnegative square root. The radicand is the number under the radical sign.

EX. in the radicand is 25 The entire expression is called a radical.

indicates the nonnegative or principal square root of 25.

indicates the negative square root of 25. is not a real number.

The following generalizations can be made:1. Every positive real number has two square roots; one is

positive and the other is negative. They are opposites of each other.

2. Negative real numbers have no real number square roots because any nonzero real number is positive when squared.

3. The square root of 0 is 0. If a 0 and b 0, then if and only if a2 = b;

a is called the principal square root of b.

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1. 2. 3. 4.

Worksheet 32 (6.2)

Summary 2:Warm-up 2. Evaluate:

a) =

To cube a number means to use the number as a factor three times.

EX. 33 = 333 = 27 and (-4)3 = (-4)(-4)(-4) = -64 A cube root of a number is one of its three equal factors.

EX. 3 is a cube root of 27 because 333 = 27-4 is a cube root of -64 because (-4)(-4)(-4) = -64

designates the cube root of a number. if and only if a3 = b;

a is called the principal cube root of b or just the cube root of b.

The concept of root can be extended to fourth roots, fifth roots, and, in general, nth roots. The following generalizations can be made:

If n is an even positive integer, then the following statements are true:

1. Every positive real number has exactly two real nth roots, one positive and one negative. 2. Negative real numbers do not have real nth roots.

If n is an odd positive integer greater than one, then the following statements are true:

1. Every real number has exactly one real nth root.2. The real nth root of a positive number is positive. 3. The real nth root of a negative number is negative.

In general, if and only if an = b. designates the principal nth root.

The n in the radical is called the index of the radical.

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b) = c) =

Worksheet 32 (6.2)

Problems - Evaluate:5. 6. 7. Summary 3:

Summary 4:

Properties of Radicals

For any nonnegative number b, if n is any positive integer greater than one, then: EX.

For any negative number b, if n is an odd positive integer greater than one, then: EX. If and are real numbers, then:

1. (The nth root of a product is equal to the product of the nth roots.)

2. (The nth root of a quotient is equal to the quotient of the nth roots.)

Simplest Radical Form

A radical is said to be in simplest radical form if the following conditions are satisfied:

1. No fraction appears under a radical sign.2. No radical appears in the denominator. 3. No radicand contains any perfect powers of the

index. 126

Warm-up 3. Change to simplest radical form:a) = b)

Worksheet 32 (6.2)

c) d)e)

Problems - Change to simplest radical form:8. 9. 10. 11. 12.

Summary 5:

Warm-up 4. Simplify:

called rationalizing the denominator. In this process the numerator and denominator of the fraction are multiplied by a number which will make the denominator become a rational number.

The process of removing the radical from the denominator of a fraction is

127

a)

b)

c)

Worksheet 32 (6.2)

d)

Note: In part d, the problem was reducible to a perfect square, therefore did not have to be rationalized.

Problems - Simplify:

13.

14.

15.

Summary 6:

K = This formula can be used to determine the area (K) of a triangle with sides a, b, c. The letter s represents the semiperimeter of the triangle. The semiperimeter is found by taking the sum of the sides, and dividing by 2.

semiperimeter = s =

Many real-world and geometric applications involve radical expressions. One which will be demonstrated here is called Heron's Formula:

128

Warm-up 5. a) Find the area of a triangle with sides of length 8, 10 and 12.

(We must first find the semiperimeter):s = K =

Worksheet 32 (6.2)

K =K =K = (Use a calculator and round to the

nearest tenth.)K =

The area of the triangle is square units.

Problems

16. Find the area of a triangular scarf with sides of 12 inches, 12 inches and 18 inches. (Round to the nearest tenth.)

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Worksheet 33 (6.3)

6.3 Combining Radicals and Simplifying Radicals That Contain Variables

Summary 1:

Warm-up 1. Simplify:a)

= =

b) =

= = - =

Only radicals with the same index and the same radicand can be added or subtracted. (These are sometimes called like radicals.)

Radical expressions can be simplified by using the distributive property to combine like radicals. Before combining radicals it is sometimes necessary to first express the radicals in simplest form.

130

= Problems - Simplify:1.

2. Worksheet

33 (6.3)

Note: To avoid the necessity for absolute value when simplifying radicals containing variables, we shall assume that all variables represent positive real numbers.

Summary 2:

Warm-up 2. Simplify:a)

= =

b) = =

Problems - Simplify:3.

4.

Summary 3:

To simplify radicals containing variables, the same basic approach is used as with numerical factors. The variables are factored using the largest possible perfect power of the index.

131

Worksheet 33 (6.3)

Warm-up 3. Simplify:a)

=

=

b)

=

=

=

=

c)

=

= Problems - Simplify:5.

Worksheet 33 (6.3)

6.

To rationalize denominators containing variables under the radical, multiply the numerator and denominator of the fraction by the same radical to make the denominator become a perfect power of the index.

132

7.

Warm-up 4. Simplify:a) =

= =

b) = = =

=

=

Problems - Simplify:8.

9.

Worksheet 34 (6.4)

6.4 Products and Quotients Involving Radicals

Summary 1:

Warm-up 1. Multiply and simplify where possible:

The property can also be used to multiply radicals and express the product in simplest form.

133

a) b) c)

= = ( )( ) =

d) = ( ) =

Problems - Multiply and simplify where possible:1. 2. 3. 4.

Summary 2:

Worksheet 34 (6.4)

Warm-up 2. Multiply and simplify where possible.a)

= =

b) = = ( )( ) =

c) =

= =

d) =

= =

e)

The distributive property can be used to find the product of radicals involving binomials.

134

=

= =

= + Worksheet 34 (6.4)

f) =

= - =

Note: The problem in part f fit the special product pattern: (a + b)(a - b) = a2 - b2. Notice that the final product contains no radical, therefore is rational. This concept is used in rationalizing binomial denominators containing radicals.Binomials such as and are called conjugates.

Problems - Multiply and simplify where possible.5.

6.

7.

8.

9.

Summary 3:

To rationalize a denominator containing a binomial radical, multiply the numerator and the denominator of the fraction by the conjugate of the denominator.

135

Worksheet 34 (6.4)

Warm-up 3. Rationalize the denominator and simplify:a)

=

=

=

=

b)

=

=

=

Problems - Rationalize the denominator and simplify:10.

11.

Worksheet 35 (6.5)

6.5 Equations Involving Radicals

Summary 1:

136

Summary 2:

Radical Equations

A radical equation is an equation containing radicals with variables in a radicand. To solve radical equations the following property is used:

Let a and b be real numbers and n be a positive integer, If a = b, then an = bn

(This property states that both sides of an equation can be raised to a positive integral power.) This process will sometimes produce answers that do not satisfy the original equation. These extra solutions are called extraneous solutions.

To Solve Radical Equations Containing One Radical

1. Isolate the radical on one side of the equation. 2. Raise both sides of the equation to the power corresponding to the index of the radical. 3. Solve the resulting linear or quadratic equation. 4. Check all solutions in the original equation for extraneous solutions.

137

Warm-up 1. Solve:a) Check:

- 0

= = 0

x = Solution Set = { }

x =

Worksheet

35 (6.5)b)

=

= 0 ( )( ) = 0

= 0 or = 0 x = or x =

Check: or

= -2 = -2

Solution Set = { }

c)

= - + 0 = - +

0 = (a - )(a - ) a - = 0 or a - = 0

a = or a = Check: or

138

=

Solution Set = { }

Problems - Solve:1.

Worksheet 35 (6.5)2.

3.

Summary 3:

To Solve Radical Equations Containing Two Radicals

1. Separate the two radicals, one on each side of the equation. 2. Raise each side of the equation to the power corresponding to the index of the radical. 3. If a radical still remains in the problem, isolate the radical. 4. Raise each side of the equation to the power corresponding to the index of the radical. 5. Solve the resulting linear or quadratic equation. 6. Check all solutions in the original equation for extraneous solutions.

139

Warm-up 2. Solve:a)

= + + = =

( )2 = =

= xWorksheet

35 (6.5)

Check:

- 2 = 2

Solution Set= { }

Problem - Solve:4.

140

Worksheet 36 (6.6)

6.6 Merging of Exponents and Roots

Summary 1:

If b is a real number, n is a positive integer greater than one and exists then: If is a rational number, where n is a positive integer greater than one and b is a real number such that exists, then:

Notice that the denominator of the exponent, n, becomes the index of the radical; and the numerator of the exponent, m, becomes the exponent of the radicand or the root.

Rational Exponents

141

Warm-up 1. Simplify:a)

b)

c)

d)

e)

f)

Worksheet 36 (6.6)

Problems - Simplify:1. 2.

3.

4.

5.

6.

Summary 2:

Problems can be written in exponential form or radical form. Sometimes it is helpful to be able to switch from one form to the other.

142

Warm-up 2. Write in radical form:a) b) c) d)

Problems - Write in radical form:7. 8. 9.

10.

Worksheet 36 (6.6)

Warm-up 3. Write using positive rational exponents:a)

b)

c)

Problems - Write using positive rational exponents:11. 12. 13.

Summary 3:

Warm-up 4. Simplify, expressing results using positive exponents only:

a)

The basic properties of exponents are used to simplify expressions containing rational exponents.

143

= = =

b)

= = =

Worksheet 36 (6.6)

c)

=

Problems - Simplify, expressing results using positive exponents only:

14.

15.

16.

Summary 4:

Some radicals can be multiplied and divided, even if they have a different index, by changing to exponential form, using the properties of exponents, and changing back to radical form.

144

Warm-up 5. Perform the indicated operations and express the answer in simplest radical form.

a) =

= = = Worksheet 36 (6.6)

b)

=

=

=

=

=

=

Problems - Perform the indicated operations and express the answer in simplest radical form:

17.

18.

Worksheet 37 (6.7)

145

6.7 Scientific Notation

Summary 1:

Summary 2:

Scientific Notation

Scientific notation or scientific form is used to more conveniently represent very small or very large numbers. To express a number in scientific notation, write the number as a product of a number between 1 and 10 and an integral power of 10. (N) (10)k

EX. (2.5) (10)3 (3.62) (10)-4

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Warm-up 1. Write in scientific notation:

a) 476 = (4.76) (10)( )

b) 0.000096456 = (9.645) (10)( )

c) 29,000,000 = ( ) (10)( )

d) 0.0136579 = ( ) (10)( )

Procedure to change from ordinary notation to scientific notation:

1. Write the number as the product of a number between 1 and 10 and a power of ten. (There should be only one digit to the left of the decimal.) 2. The exponent of the 10 corresponds to the number of places the decimal point moves when changing the number to a number between 1 and 10. 3. The exponent is:

- negative if the original number is less than 1: 0.0036 = (3.6) (10)-3

(Decimal moved to the right.)- positive if the original number is greater than 1: 563 =

(5.63) (10)2

(Decimal moved to the left.)- 0 if the original number is between 1 and 10: 9.38 =

(9.38) (10)0

(Decimal did not move.)

147

Worksheet 37 (6.7)

Problems - Write in scientific notation:

1. 5,611

2. 0.000319

3. 520,000,000,000

4. 0.000000872

5. 3.51267

Summary 3:

Warm-up 2. Write in ordinary notation:

a) (9.345) (10)5 = (Decimal moves 5 places to the right.)

b) (2.15) (10)-4 = (Decimal moves 4 places to the left.)

1. Move the decimal point the number of places indicated by the exponent of the 10. 2. The decimal point moves to the right if the exponent is positive: EX. (3.19) (10)3 = 3,190 3. The decimal point moves to the left if the exponent is negative: EX. (2.793) (10)-2 = 0.02793

Procedure to change from scientific notation to ordinary notation:

148

c) (6) (10)3 =

d) (2) (10)-4 =

Problems - Write in ordinary notation:

6. (2.49678) (10)6

7. (9) (10)-2

8. (3.1864) (10)-3

9. (7) (10)4

Worksheet 37 (6.7)

Summary 4:

Warm-up 3. Perform the indicated operations using scientific notation and the properties of exponents:

a) (25,600) (1,900) = (2.56) (10)4 (1.9) (10)3

= (2.56) (1.9) (10)4 (10)3

= ( ) (10)( )

=

b) (0.000341) (0.0019) = ( ) (10)( ) ( ) (10)( )

= ( ) ( ) (10)( ) (10)( )

= ( ) (10)( )

=

c)

To simplify numerical calculations, change the numbers to scientific notation and use the properties of exponents.

149

= ( ) (10)( )

= d)

=

=

=

Worksheet 37 (6.7)

Problems - Perform the indicated operations using scientific notation and properties of exponents:

10. (8,942) (1,500,000)

11. (0.00025) (0.006)

12.

13.

Summary 5:

150

Scientific Notation and Calculators Most calculators will automatically display answers in scientific notation if the answer exceeds the calculator's display capabilities.

420,0002 will be displayed as 1.764 E11 on a graphing calculator

420,0002 will be displayed as 1.764 11 on most scientific calculators.

Many calculators also have a key to select scientific notation mode. When this is selected, all answers will be displayed in scientific notation.

Many calculators also allow you to enter a number in scientific notation by using an enter-the-exponent key. (EE or E EX or EXP)

It is important that you have a thorough understanding of the capabilities of your calculator.

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