Grade 7- Review on Bases and Exponents

2. :
In this lesson, the students must be able to:
1) define and interpret the meaning of an where n is a positive integer;
2) derive inductively the Laws of Exponents (restricted to positive
integers)
3) illustrate the Laws of Exponents.
2

3. Activity 1:
Give the product of each of the following as fast as you can.
3
91) 3 x 3 = ________
2) 4 x 4 x 4 = ________
3) 5 x 5 x 5 = ________
4) 2 x 2 x 2 = ________
5) 2 x 2 x 2 x 2 = ________
6) 2 x 2 x 2 x 2 x 2 =_______
64
125
8
16
32

4. an = a • a • a • a ….. (n times)
4
Correct
1. Which of the following is/are correct?
a) 42 = 4 • 4 = 16
b) 24 = 2 • 2 • 2 • 2 = 8
c) 25 = 2 • 5 = 10
d) 33 = 3 • 3 • 3 = 27
Wrong
Wrong
Correct
In an, a is called the base and n is called the exponent.

5. an = a • a • a • a ….. (n times)
In an, a is called the base and n is called the exponent.
5
8
2) Give the value of each of the following as fast as you can.
a) 23 =
b) 25 =
c) 34 =
d) 106 =
32
81
1,000,000

6. an = a • a • a • a ….. (n times)
In an, a is called the base and n is called the exponent.
6
8
2) Give the value of each of the following as fast as you can.
a) 23 =
b) 25 =
c) 34 =
d) 106 =
32
81
1,000,000

7. Activity 2
Evaluate the following. Investigate the result. Make a simple
conjecture on it.The first two are done for you.
1) (23)2 =
2) (x4)3 =
3) (32)2 =
4) (22)3 =
32 • 32
22 • 22 • 22
23 • 23 = 2 • 2 • 2 • 2 • 2 • 2 = 64
x4 • x4 • x4
= x • x • x • x • x • x • x • x • x • x • x • x = x12
= 3 • 3 • 3 • 3 = 81
= 2 • 2 • 2 • 2 • 2 • 2 = 64

8. Activity 2
Evaluate the following. Investigate the result. Make a simple
conjecture on it.The first two are done for you.
5) (a2)5=
What can you conclude about (an)m?
What will you do with a, n and m?
7) ( y12 )5
6) ( x100 )3
a2 • a2 • a2 • a2 • a2 = a10
Copy the base (a) then multiply
the exponents. ( n and m )
= (y12•5) = y60
= (x100•3) = x300
( a ) n•m = ( an•m )

9. Activity 3
Evaluate the following. Notice that the bases are the same.
The first example is done for you.
1) (23) (22) =
2) (x5) (x4) =
3) (32) (34) =
4) (24) (25) =
5) (x3) (x4) =
= 36 = 729
= 29 = 512
= 25 = 322 • 2 • 2 • 2 • 2
= x9x • x • x • x • x • x • x • x • x
3 • 3 • 3 • 3 • 3 • 3
2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2
= x7x • x • x • x • x • x • x

10. What can you conclude about (an)•(am)?
What will you do with a, n and m?
7) ( y59 ) • ( y51 )
6) ( x32 ) • ( x25 )
Copy the base (a) then add
the exponents. ( n and m )
= (y59+51) = y110
= (x32+25) = x57
( a ) n+m = ( an+m )

11. Activity 4
Evaluate each of the following. Notice that the bases are the
same.The first example is done for you.
1)
27
23 =
2)
35
33 =
3)
43
42 =
4)
28
26 =
=
128
8
2•2•2•2•2•2•2
2•2•2
= 24
= 16
3•3•3•3•3
3•3•3
=
243
27
= 9 = 32
4•4•4
4•4
=
64
16
= 4 = 41 = 4
2•2•2•2•2•2•2•2
2•2•2•2•2•2
=
256
64
= 4 = 22

12. What can you conclude about
𝑎 𝑛
𝑎 𝑚 ?
What will you do with a, n and m?
7)
𝑦105
𝑦87
6)
𝑥20
𝑥13
Copy the base (a) then subtract
the exponents. ( n and m )
= (y105-87) = y18
= (x20-13) = x7
( a ) n ─ m = ( an ─ m )

13. Summary
Laws of Exponent
1) an = a • a • a • a • a….. (n times)
2) (an)m = an•m Power of powers
3) an • am = a m + n Product of a power
4)
𝑎 𝑛
𝑎 𝑚
5) a0 = 1 where a ≠ 0 Law for zero exponent
Quotient of power= a n ─ m

14. 5. Law for Zero Exponent
a0 = 1 where a ≠ 0
What about these?
a) (7,654,321)0
b) 30 + x0 + (3y)0
= 1
= 1 + 1 + 1
= 3

15. 6. Law for Negative Exponent
a ─ n =
1
𝑎 𝑛 ; where a ≠ 0
Rewrite the fractions below using exponents and simplify them?
a.
2
4
=
21
22
b.
4
32
=
22
25
c.
27
81
=
33
34
or
2
4
=
1
2
= 2─1 =
1
21
= 8─1 =
1
81 or 2─3 =
1
23
= 3─1 =
1
31
or
4
32
=
1
8
or
27
81
=
1
3

16. Laws of Negative Exponent
What about these?
d) x─ 2
e) 3 ─ 3
f) (5 ─ 3) ─ 2
=
1
27
=
1
33
=
1
𝑥2
=
1
22= 2 ─ 2 =
1
4