Name:___________________________
Exponents and Scientific Notation Review
Date:______________
8th Grade Mathematics
Basic:


Multiplication is repeated addition
o That means:
 Numerically: 5 + 5 + 5 + 5 + 5 + 5 can be written as 5 x 6
 Algebraically: x + x + x + x + x + x + x + x can be written as 8x
Exponents are repeated multiplication
o That means:
 Numerically: 5 x 5 x 5 x 5 x 5 x 5 can be written as 56
 Algebraically: x ∙ x ∙ x ∙ x ∙ x ∙ x ∙ x ∙ x can be written as x8
Exponential Notation
Notes:



Base: a number being raised to a power
Exponent: tells how many times the base is used as a factor
Power: a number written in exponential notation
Practice:
1. When a negative number is raised to an odd power, what is the sign of the result?
2. When a negative number is raised to an even power, what is the sign of the result?
3. Why should we bother with exponential notation? Why not just write out the multiplication?
4. Rewrite each of the following in exponential notation:
a. 12 ∙ 12
b. 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8
c. 15 ∙ 15 ∙ 15 ∙ 15 ∙ 15 ∙ 15 ∙ 15
d. 7 ∙ 7 ∙ 7 ∙ 7
e. r ∙ r ∙ r ∙ r ∙ r
f.
b∙b∙b
g. n ∙ n ∙ n ∙ n ∙ n ∙ n ∙ n ∙ n ∙ n ∙ n ∙ n ∙ n
5. What is the difference between the following two expressions: 56 and 5(6)?
6. Sophia thinks that 64 can be written in exponential notation using 2, 4, or 8 as the base. Is she correct? Explain.
7. Rewrite each number in exponential notation using 2 as the base.
a. 1
b. 2
c. 4
d. 8
e. 16
f.
32
g. 64
h. 128
i.
256
j.
512
8. Rewrite each number in exponential notation using 3 as the base.
a. 1
b. 3
c. 9
d. 27
e. 81
f.
243
g. 729
h. 2187
i.
6561
j.
19683
Multiplication of Numbers in Exponential Form
Notes:



When multiplying powers of the same base, add the exponents.
Only add exponents if the powers are being multiplied and they contain the exact same base.
Remember: bases that contain fractions and/or negative numbers require parentheses.
Notes:



When dividing powers of the same base, subtract the exponents.
Only subtract exponents if the powers are being divided and they contain the exact same base.
Remember: bases that contain fractions and/or negative numbers require parentheses.
Practice:
1. Using your knowledge of exponents demonstrate why 22 ∙ 42 = 26.
2. Simplify each of the following expressions:
a. 127 ∙ 125
b. x6 ∙ x9
c. 45 ∙ 46 ∙ 49
d. 32 ∙ 43 ∙ 9
e. 24 ∙ 43 ∙ 82 ∙ 28
f.
x y xy 
g.
78
7
h.
46
23
i.
x9
x3
j.
xy 5
xy 3
3
4
5
3. A rectangular area of land is being sold off in smaller pieces. The total area of the land is 39 square miles. The
pieces that are being sold are 92 square miles in size. How many smaller pieces of land can be sold at the stated
size? Compute the actual number of pieces.
4. The dimensions of a rectangular prism are 8xy5 inches long, 7x2y3 wide and 3x4y6 inches high. What is the
volume of the prism expressed in simplest form?
5. The area of a rectangle is 64x5y8 inches squared. If the length of the rectangle is 23x2y5 inches, what is the
width?
Numbers in Exponential Form Raised to a Power
Notes:

When raising a power to a power, multiply the exponents.
Notes:

If the term in the parentheses contains multiple parts (number and a variable/s), each part is raised to the
exponent of the parentheses
o

4 x y 
x  y n
2
3 5
 1024 x10 y 15
 xn  yn
Practice:
1. How is  xy related to x n and y n ?
n
n
x
2. How is   related to x n and y n , y  0 ?
 y
3. Christopher wrote that (52)3 = 55. Correct his mistake. Write an exponential expression using a base of 5 and
exponents of 2, 3, and 5 that would make his answer correct.
4. Show (prove)  xyz  x 3 y 3 z 3 for any numbers x, y, and z.
3
5. Simplify each of the following expressions:
a.
4 
b.
x 
c.
2 
d.
3x 
e.
5x y 
3 5
4 9
4 2
3 3
3
4 2
Numbers Raised to the Power of Zero

Any number raised to the power of zero is one.
Practice:
 54  0
1. Simplify the following expression as much as possible:  4   6
5 
2. Let x and y be two numbers. Use the distributive law and the definition of the zeroth power to show that the
𝟎
𝟎
𝟎
numbers (𝒂𝟎 + 𝒃 ) 𝒂𝟎 and (𝒂𝟎 + 𝒃 ) 𝒃 are equal.
Negative Exponents and the Laws of Exponents
Notes:


Negative exponents are the reciprocal of the power.
A number multiplied by its reciprocal is the inverse.
Practice:
1. Write the complete expanded form of the number 896,348 in exponential notation.
2. Write the complete expanded form of the decimal 6.2357 in exponential notation.
𝟕 −𝟒
3. Show directly that (𝟓)
=
𝑎
𝟕−𝟒
𝟓−𝟒
.
4. If we let 𝒃 = −𝟏 in (𝑥 𝑏 ) = 𝑥 𝑎𝑏 , 𝒂 be any integer, and 𝒚 be any positive number, what do we get?
Laws of Exponents
For any numbers 𝑥, 𝑦 (𝑥 ≠ 0 in (4) and 𝑦 ≠ 0 in (5)) and any positive integers 𝑚, 𝑛, the following holds:
𝑥 𝑚 ∙ 𝑥 𝑛 = 𝑥 𝑚+𝑛
(1)
(𝑥 𝑚 )𝑛 = 𝑥 𝑚𝑛
(2)
(𝑥𝑦)𝑛 = 𝑥 𝑛 𝑦 𝑛
(3)
𝑥𝑚
= 𝑥 𝑚−𝑛
𝑥𝑛
(4)
𝑥 𝑛 𝑥𝑛
( ) = 𝑛
𝑦
𝑦
(5)
For any numbers 𝑥, 𝑦 and for all whole numbers 𝑚, 𝑛, the following holds:
𝑥 𝑚 ∙ 𝑥 𝑛 = 𝑥 𝑚+𝑛
(6)
(𝑥 𝑚 )𝑛 = 𝑥 𝑚𝑛
(7)
(𝑥𝑦)𝑛 = 𝑥 𝑛 𝑦 𝑛
(8)
For any positive number 𝑥 and all integers 𝑏
𝑥 −𝑏 =
1
𝑥𝑏
(9)
For any numbers 𝑥, 𝑦 and all integers 𝑎, 𝑏, the following holds:
𝑥 𝑎 ∙ 𝑥 𝑏 = 𝑥 𝑎+𝑏
(10)
(𝑥 𝑏 )𝑎 = 𝑥 𝑎𝑏
(11)
(𝑥𝑦)𝑎 = 𝑥 𝑎 𝑦 𝑎
(12)
𝑥𝑎
= 𝑥 𝑎−𝑏
𝑥𝑏
(13)
𝑥 𝑎 𝑥𝑎
( ) = 𝑎
𝑦
𝑦
(14)