Law of exponents

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} = 2 ⁸

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} = 5 ⁹

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}       (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

  (8 ⁹ ) 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) ²

∴ 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