We know how to calculate the expression 5 x 5. This expression can be written in a shorter way using something called exponents.
5⋅5=52
An expression that represents repeated multiplication of the same factor is called a power.
The number 5 is called the base, and the number 2 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.
figure10
31 3 to the first power 3
42 4 to the second power or 4 squared 4 ∙ 4
53 5 to the third power or 5 cubed 5 ∙ 5 ∙ 5
26 2 to the power of six 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2
Example
Write these multiplications like exponents
5⋅5⋅5=53
4⋅4⋅4⋅4⋅4=45
3⋅3⋅3⋅3=34
Multiplication
If two powers have the same base then we can multiply the powers. When we multiply two powers we add their exponents.
The rule:
xa⋅xb=xa+b
Example
42⋅45=(4⋅4)⋅(4⋅4⋅4⋅4⋅4)=47=42+5
Division
If two powers have the same base then we can divide the powers. When we divide powers we subtract their exponents.
The rule:
xaxb=xa−b
Example
4245=⧸4⋅⧸4⧸4⋅⧸4⋅4⋅4⋅4=143=4−3=42−5
A negative exponent is the same as the reciprocal of the positive exponent.
x−a=1xa
Example
2−3=123
When you raise a product to a power you raise each factor with a power
(x⋅y)a=xa⋅ya
Example
(2x)4=24⋅x4=16x4
The rule for the power of a power and the power of a product can be combined into the following rule:
Answers & Comments
Answer:
Powers and exponents
We know how to calculate the expression 5 x 5. This expression can be written in a shorter way using something called exponents.
5⋅5=52
An expression that represents repeated multiplication of the same factor is called a power.
The number 5 is called the base, and the number 2 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.
figure10
31 3 to the first power 3
42 4 to the second power or 4 squared 4 ∙ 4
53 5 to the third power or 5 cubed 5 ∙ 5 ∙ 5
26 2 to the power of six 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2
Example
Write these multiplications like exponents
5⋅5⋅5=53
4⋅4⋅4⋅4⋅4=45
3⋅3⋅3⋅3=34
Multiplication
If two powers have the same base then we can multiply the powers. When we multiply two powers we add their exponents.
The rule:
xa⋅xb=xa+b
Example
42⋅45=(4⋅4)⋅(4⋅4⋅4⋅4⋅4)=47=42+5
Division
If two powers have the same base then we can divide the powers. When we divide powers we subtract their exponents.
The rule:
xaxb=xa−b
Example
4245=⧸4⋅⧸4⧸4⋅⧸4⋅4⋅4⋅4=143=4−3=42−5
A negative exponent is the same as the reciprocal of the positive exponent.
x−a=1xa
Example
2−3=123
When you raise a product to a power you raise each factor with a power
(x⋅y)a=xa⋅ya
Example
(2x)4=24⋅x4=16x4
The rule for the power of a power and the power of a product can be combined into the following rule:
(xa⋅yb)z=xa⋅z⋅yb⋅z
Example
(x3⋅y4)2=x3⋅2⋅y4⋅2=x6⋅y8
Video lessons
Rewrite the expressions
2⋅2⋅2
x⋅x⋅x⋅x⋅x
34
x3