##
**
Multiplying numbers with same base
**

a
^{
m
}
× a
^{
n
}
= a
^{
m + n
}

Let’s take some examples & check

2
^{
5
}
× 2
^{
3
}
= (2 × 2 × 2 × 2 × 2) × (2 × 2 × 2)

= 2
^{
8
}

∴ 2
^{
5
}
× 2
^{
3
}
= 2
^{
5 + 3
}
= 28

Similarly,

(−3)
^{
4
}
× (−3)
^{
2
}

= (−3 × −3 × −3 × −3) × (−3 × −3)

= −3 × −3 × −3 × −3 × −3 × −3

= (−3)
^{
6
}

∴ (−3)
^{
4
}
(−3)
^{
2
}
= (−3)
^{
4 + 2
}
= (−3)
^{
6
}

And for,

5
^{
6
}
× 5
^{
3
}
= (5 × 5 × 5 × 5 × 5 × 5) × (5 × 5 × 5)

= 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5

= 5
^{
9
}

∴ 5
^{
6
}
× 5
^{
3
}
= 5
^{
6
}
^{
+ 3
}
= 59

**
Dividing Numbers with same base
**

a
^{
m
}
/a
^{
n
}
= a
^{
m - n
}

2
^{
5
}
/2
^{
3
}
= (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2)

= 2 × 2

= 2
^{
2
}

∴ 2
^{
5
}
/2
^{
3
}
= 2
^{
5 - 3
}
= 2
^{
2
}

Similarly,

-3
^{
4
}
/-3
^{
2
}
= (-3 × -3 × -3 × -3)/(-3 × -3)

= -3 × -3

= (-3)
^{
2
}

∴ (-3)
^{
4
}
/(-3)
^{
2
}
= (-3)
^{
4 - 2
}
= (-3)
^{
2
}

And for,

5
^{
6
}
/5
^{
3
}
= (5 × 5 × 5 × 5 × 5 × 5)/(5 × 5 × 5)

= 5 × 5 × 5

= 5
^{
3
}

∴ 5
^{
6
}
/5
^{
3
}
= 5
^{
6
}
^{
− 3
}
= 5
^{
3
}

###
**
Power of a Power
**

Thus we can write

(a
^{
m
}
)
^{
n
}
= a
^{
m × n
}

Suppose we have

(2
^{
3
}
)
^{
2
}
= 2
^{
3
}
× 2
^{
2
}

= 2
^{
3
}
× 2 (As a
^{
m
}
× a
^{
n
}
= a
^{
m + n
}
)

= 2
^{
6
}

So, (2
^{
3
}
)
^{
2
}
= 2
^{
3
}
× 2 = 2
^{
6
}

Similarly,

(3
^{
2
}
)4 = 3
^{
2
}
× 4

= 3
^{
8
}

(7
^{
4
}
)
^{
5
}
= 7
^{
4 × 5
}

= 7
^{
20
}

(89)
^{
3
}
= 8
^{
9 × 3
}

= 8
^{
27
}

###
**
Multiplying number with same power
**

We note that

a
^{
m
}
× b
^{
m
}
= (a × b)
^{
m
}

Suppose we have

2
^{
2
}
× 3
^{
2
}
= (2 × 2) × (3 × 3)

= (2 × 3) × (2 × 3)

= 6 × 6

= 6
^{
2
}

Thus,

2
^{
2
}
× 3
^{
2
}
= (2 × 3)
^{
2
}

= 6
^{
2
}

Similarly,

5
^{
3
}
× 7
^{
3
}

= (5 × 5 × 5) × (7 × 7 × 7)

= (5 × 7) × (5 × 7) × (5 × 7)

= 35 × 35 × 35

= 35
^{
3
}

∴ 5
^{
3
}
× 7
^{
3
}
= (5 × 7)
^{
3
}

= 35
^{
3
}

###
**
Dividing number with same power
**

We note that

a
^{
m
}
/b
^{
m
}
= (a/b)
^{
m
}

Suppose we have,

2
^{
2
}
/3
^{
2
}
= (2 × 2)/(3 × 3)

= 2/3 × 2/3

= (2/3)
^{
3
}

**
∴ 2
^{
2
}
/3
^{
2
}
= (2/3)
^{
2
}
**

Similarly,

5
^{
3
}
/7
^{
3
}
= (5 × 5 × 5)/(7 × 7 × 7)

= (5/7)×(5/7)×(5/7)

= (5/7)
^{
3
}

**
∴ 5
^{
3
}
/7
^{
3
}
= (5/7)
^{
3
}
**