Notes Booklet
Exponents 9
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Exponents
  • Exponents are a form of writing an expression when the factors are used

24

  • spoken:
  • expanded (what it means):
  • standard form (answer):

power / base / exponent / expanded form / standard form
82
10 • 10 • 10 • 10 • 10 • 10
3 / 4
4 / 3
  • either the base or the exponent can be written with


Negative Exponents

5–3

  • when the is negative, use the of the base to a exponent

7   10

Exponent / Expanded
Form / Standard
Form
Negative / Positive
5–3 / ()3 / • • /
2–8
()–3
()–6
  • when working in fractions, leave your answer as a fraction

Negative Bases
  • when the is negative, follow the (B – E/R – D/M – A/S)

(–5)4(–5) • (–5) • (–5) • (–5) + 625

–54– ( 5 • 5 • 5 • 5 )– 625

Integral Exponents
  • integral exponents are of , either or

Positive Integral Exponents
Standard / Expanded / Exponent
1 000 000
1000
100
10
1
Negative Integral Exponents
negative exponent / integral fraction / fraction / decimal
10-4
  1. 000 000 1

Place Values and Integral Exponents

987 654 321 .123 456 789

Exponent Laws

First power law
  • any number can be expressed as to the power

17 = – =

Multiplying exponents law
  • when 2 similar exponent expressions are multiplied, the exponents are

• = =

proof:

Dividing exponents law
  • When 2 similar exponent expressions are divided, the exponents are

÷ = =

proof:

Exponents to an exponent
  • when an exponent expression is raised to the exponents are

= =

proof:

Zero power
  • any number to the power is always

= 1 = 1 = 1

Proof:

Negative exponents
  • when a exponent is used, it can be expressed as the using a exponent

=

Powers involving fractions as the base
  • when fractions are used as the base, the and the are raised to the power

=

Scientific Notation

5 906 000 000 kmdistance between the Sun and Pluto

0.000 000 3 msize of mycoplasma bacteria

  • Scientific notation is a form of expressing very or very numbers
  • it uses and of 10.
Steps to Follow

15 000 000 000 years

  1. move the to create a between

15 000 000 000 1.5

  1. add the base

15 000 000 000 1.5 • 10

  1. the number of moved to form the exponent

15 000 000 000 1.5 • 1010

decimal moved to the left = positive exponent

decimal moved to the right = negative exponent

Writing Standard Form From Scientific Notation

1.4 • 1019kmdistance to the nearest galaxy

1.0 • 10–16 mdiameter of a proton

1.move the the number of places shown by the

positive exponent = decimal moved to the right ()

negative exponent = decimal moved to the left ()

2. the base ( • 10)

3.use to any empty place values

1.4 • 1019 

1.0 • 10–16

Practicing Scientific Notation
  • Express the following in scientific notation:
  1. 9700
  1. 59 400 000
  1. 76 000 000 000
  1. 0.000 4
  1. 0.000 005 73
  • Express the following in standard form:
  1. 6.55 • 104
  1. 9.01 • 1011
  1. 6.55 • 10-8
  1. 1.1 • 10-4
  1. 3.027 • 10–8

We Could Put More Notes Here (If We Want)


Scientific Notation and Place Values 1

(6 • 104) + (2 • 102) + (3 • 100) + (7 • 10–3)

1.your place values

2.place the into the correct place value

3.the remaining place values with

Scientific Notation and Place Values 2

30 217.08

1.identify the correct for each digit

30 217.08

2.place the digit with the corresponding exponent

(3 •) + (2 •) + (1 •) + (7 •) + (8 •)

3.use to separate each place value

Operations With Scientific Notation 1
  1. convert all values into

=

  1. the operations together
  1. the together

() • ()

  1. / the as needed
  1. use the to simplify the exponents

4 •4 •

Operations With Scientific Notation 2

4000 000 • 0.000 008

  1. convert all values into

(4.0• 107) • (8.0 • 10–6)

  1. the operations together
  1. the together

(4.0 • 8.0) • (107• 10–6)

  1. / the as needed
  1. use the to simplify the exponents

32•32•

  1. the into scientific notation
  1. the

(3.2 • 101) • 1013.2 • 102

Scientific Notation Practice
We Could Put More Notes Here (If We Want)
Rational Numbers

  • rational numbers are numbers that can be written as the of 2 integers or polynomials
  • irrational numbers be expressed as a

Rational Numbers / Irrational Numbers
Square Roots
  • the square root of a number is the value that, when multiplied by gives the number
  • exponents and roots are operations

= 552 (5 • 5)= 25

Understanding Square Roots
  • the square root represents the of its square

< <

  • is a number that can be
  • no matter how we make each side, there will always be a piece
  • it is a decimal and is therefore

Cube Roots
  • the cube root of a number is the value that, when multiplied by gives the number
  • exponents and roots are operations

= 553 (5 • 5 • 5)= 125

Understanding Cube Roots
  • the square root represents the of its cube

< <

Perfect Squares

1491625

36496481100

121144169196225

256289324361400

Perfect Cubes

182764125

2163435127291000

Square Roots of Variables
  • the square root of a number is the value that, when multiplied by gives the number

n2 = n • n  =

 = n

n10 = n • n • n • n • n • n • n • n • n • n

=

=

= n5

  1. the remains the
  1. the exponent by

Order of Operations

4 + 6(1 + 32) ÷ 4 +

  • when there are operational signs (+, – , •, ÷, brackets, exponents) within an equation, there is a specific to follow

BEDMAS BERDMAS

B:

E: / R

(working from left to right)

D: / M:

(working from left to right)

A: / S:

(working form left to right)

4 + 6(1 + 32) ÷ 4 +