Bell Work

Simplify by adding like terms.

4 + 7mxy + 5 + 3yxm – 15

Answer:

-6 + 10mxy

Lesson 19:Exponents, Powers of

Negative Numbers, Roots, Evaluation of

Powers

Exponents:

Exponential Notation*: The general form of the expression is x , which indicates that x is be used as a factor n times and is read “x to the nth.”

n

In this definition, the letter x represents a real number and is called the base of the expression. The letter n represents a positive integer and is called the exponent.

Power*: The value of an exponential expression.

2 = (2)(2)(2)(2) = 16

The value of 2 used as a factor four times is 16. We say that the fourth power of 2 is 16. We could also say 2 raised to the 4th power.

4

If something is raised to the 2nd power we say it is squared.

If something is raised to the 3rd power we say it is cubed.

Everything greater than 3 we say the nth power.

Powers of negative numbers:

When a positive number is raised to a positive power, the result is always a positive number.

Example:

Simplify

3 3 3 -3

2 3 4 4

Answer:

(3)(3) = 9

(3)(3)(3) = 27

(3)(3)(3)(3) = 81

Be careful because -3 means the opposite of 3 and not (-3)

-(3)(3)(3)(3) = -81

4

44

When a negative number is raised to an even power, the result is always positive; and when a negative number is raised to an odd power, the result is always negative.

Example:

Simplify

(-3) (-3) (-3) -(-3)2 3 4 4

Answer:

(-3)(-3) = 9

(-3)(-3)(-3) = -27

(-3)(-3)(-3)(-3) = 81

-(-3)(-3)(-3)(-3) = -81

Example:

Simplify

-3 – (-3) + (-2)3 2 2

Answer:

-(3)(3)(3) – (-3)(-3) + (-2)(-2)

= -27 – 9 + 4

= -32

Example:

Simplify

-2 – 4(-3) – 2(-2) – 2 2 3 2

Answer:

-(2)(2) – 4[(-3)(-3)(-3)] – 2[(-2)(-2)] – 2

= -4 – 4(-27) – 2(4) – 2

= -4 + 108 – 8 – 2

= 94

Roots:

If we use 3 as a factor twice, the result is 9. Thus, 3 is the positive square root of 9. we use a radical sign to indicate the root of a number.

(3)(3) = 9 so √9 = 3

If we use 3 as a factor three times, the result is 27. thus, 3 is the positive cube root of 27.

(3)(3)(3) = 27 so √27 = 3

3

If we use 3 as a factor four times, the result is 81. Thus, 3 is the positive fourth root of 81.

(3)(3)(3)(3) = 81 so √81 = 3

4

If we use 3 as a factor five times, the result is 243. Thus, 3 is the positive fifth root of 243.

(3)(3)(3)(3)(3) = 243 so √243 = 3

5

Because (-3)(-3) = +9, we say that -3 is the negative square root of 9. Because (-3)(-3)(-3)(-3) equals +81, we say that -3 is the negative fourth root of 81.

If n is an even number, every positive real number has a positive nth root and a negative nth root. We use the radical sign to designate the positive even root. To designate a negative even root, we also use a minus sign.

Example:

Simplify

√9 -√9 √81-√81

4 4

Answer:

√9 = 3

-√9 = -3

√81 = 3

-√81 = -3

4

4

The number under the radical sign is called the radicand, and the little number that designates the root is called the index.

Practice:

√64 √16 √-27 -√81

4 3

Answer:

√64 = 8

√16 = 2

√-27 = -3

-√81 = -9

4

3

Evaluation of Powers:

Evaluate: yx m

If y = 3, x = 4, and m = 2

(3)(4) (2) = (3)(16)(8)

=384

2 3

2 3

We must be careful, however, when the expression contain minus signs or when some replacement values of the variables are negative numbers.

If a = -2, what is the value of each of the following?

a -a -a (-a)

2 2 3 3

Answer: a = -2

a = (-2)(-2) = +4

-a = -(-2)(-2) = -4

-a = -(-2)(-2)(-2) = +8

(-a) = (-(-2))(-(-2))(-(-2)) = +8

2

2

3

3

Practice:

Evaluate

pm – z

If p = 1, m = -4, z = -2

2 3

Answer:

(1)(-4)(-4) – (-2)(-2)(-2)

= 16 + 8

= 24

HW: Lesson 19 #1-30

Due Tomorrow