exponent and exponents as repeated multiplication

1. Exponents

2. 𝑎𝑐
An exponent is a symbol or a
number at the upper right hand
corner of a variable or a constant. It
indicates the number of times the
base is used as a factor.
The base is the repeated factor in a
product.
‹— exponent
‹— base

3. An exponent tells how many times a number,
called the base, is used as a factor. Thus,
3 • 3 • 3 • 3 • 3 can be written as 35
7 • 7 • 7 • 7 • 7 • 7 • 7 can be written as 77
a • a • a • a • a • a can be written as 𝑎6
xy • xy • xy • xy can be written as (𝑥𝑦)4 or 𝑥4𝑦4
In the given exponential notations above; 3, 7, a, and xy
are bases; and 5, 7, 6, , and 4 are the exponents.

4. Exponents as
Repeated
Multiplication

5. This is the Bar of 5, and this is the Square of 5.
Five to the second power or 5 ‘squared’. We write as
52
This term was first applied to the power of 2 by al-
Khwarizmi, the Father of Algebra in the 9th century.
Euclid, a Greek
mathematician who
lived over 2000
years ago, was the
first to use the word
power. He thought
of the secondary
power as being the
square of a line.

6. The cube of 5
We write this as 53, but instead of
calling this the ‘third power,’ he
also called it ‘5 cubed.’
The words in Arabic were ‘mai’
for ‘square, and ‘kab’ for ‘cube.’
About 600 years later, other Arab
mathematicians shortened this to
just the first letter. So if they
wanted to write 53, they would
write 5k and 52 would be 5m.

7. The idea of a special notation for repeated
multiplication was very limited until the 15th
century when Nicolas Chuquet used a form of
exponents.
He didn’t invent the words million and billion, but
he was the first to use the ending ‘illion’ to
represent powers of 10. Robert Recorde invented
words for higher powers above the cube about a
hundred years later, but he is more famous for
inventing the equal sign.

8. For example, if we wanted to write one million, we could
simply write 1,000,000, which is pretty quick. But how
else could we write this? We could write 106.
10100
A number with 100 zeros. How long would it take to write
this whole number out?

9. Robert Recorde gave names to powers up to 8, but never
up to 100! The only reason this number is interestingly is
because it has exactly 100 zeros. As recently as 1938,
American mathematician Edward Kasher popularized the
name ‘googol’ for the number, although actually it was his
9 year-old nephew who called it that; and that name has
stuck ever since. So 10100 is a googol. Now let’s try and
write 10googol. This means a one followed by a googol of
zeros, and it is called a googolplex.

10. Exponent is a simple way to write a repeated
multiplication.
7.7. 7.7. 7.7. 7.7. 7.7 can be written very concisely
as 710
The number that is being repeated is called the
base, and the number at the top tells us how
many instances of the base have been multiplied
together. That number is called exponent.

11. It comes from the Latin exponere
meaning to put forth.
We can write numbers in exponential
form, like 710, or in expanded form, like
7.7. 7.7. 7.7. 7.7. 7.7.

12. Example:
Evaluate 23, 34, and 45
Solution:
23
= 2 . 2 . 2 = 8
34
= 3 . 3 . 3 . 3 = 81
45
= 4 . 4 . 4 . 4 . 4 = 1,024

13. How would we evaluate -24 ?
Solution:
-24 = - 2 . 2 . 2 . 2 = -16

14. How would we write -2 . -2 . -2 . -2?
Solution:
We enclose the “-2” inside the parentheses and raise it to
the power of 4.
-2 . -2 . -2 . -2 = −2 4
= 16