**First Law of Exponents:**

a^{m} × a^{n} = a^{ m + n}

Example 1:

3^{2} × 3^{3} = 3^{2 + 3} = 3^{5} = 243

Example 2:

2^{4} × 2^{4} = 2^{8} = 256

** Second Law of Exponents:**

a^{m} / a^{n} = a^{m – n}

Example:

3^{6}/3^{2} = 3^{6 – 2} = 3^{4} = 81

**Third Law of Exponents:**

If a is any number except 0, then

**a ^{0} = 1. **

Example 1:

5^{0} = 1

x^{0} = 1, if x is any number except 0.

Example 2:

9^{3}/9^{3} = 9^{3 – 3} = 9^{0} = 1

**Fourth Law of Exponents:**

(a^{m})^{ n} = (a)^{m × n}

Example 1:

(2^{2})^{3} = (2)^{2 × 3} = 2^{ 6} = 64

**Fifth Law of Exponents:**

(ab)^{m} = a^{m} × b^{m}

Example 1:

(10)^{5} = (2 × 5)^{5} = 2^{5} × 5^{5}

**Sixth Law of Exponents:**

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

Example:

(3/5)^{3} = 3^{3} / 5^{3} = 27/125

Very Important rules on exponents:

**1. (a) ^{?m} = 1/a^{m}**

Example:

(2) ^{?4} = 1/2^{4} = 1/16

**2. (1/a) ^{ ?m} = 1/ (1/a)^{ m} = a^{m}**

Example:

(1/3)

Example:

(2/3)

**4. (a ^{m}) ^{1/n} = (a^{m/n} )**

Example:

(2

nth root of a number Or A Surd:

Let a be a positive number and n a positive integer.

Then the nth root of a is denoted as:

nva or (a) ^{1/n}

a^{1/n} is called a Surd of order n.

Examples:

1. the 2^{nd} root of 25 is denoted as ^{2}√25.

And (25)^{1/2} = (5^{2})^{1/2} = 5^{2×1/2} = 5^{2×1/2} = 5

2. the 3^{rd} root of 27 is ^{3}√27

And (27)^{1/3} = (27)^{1/3} = (3^{3})^{1/3} = 3^{3×1/3} = 3^{3×1/3} = 3

3. The 4^{th} root of 16 is ^{4}√16

And ^{4}√16 = (16)^{1/4} = (2^{4})^{1/4} = 2^{4×1/4} = 2^{4×1/4} = 2

4. The m^{th} root of a^{m} is ^{m}√a^{m} = (a^{m})^{1/m} = a^{m×1/m} = a

nth root of a negative number:

Let a be a negative number, then the nth root of a will exist only if n is a positive odd integer, not when n is a positive even integer.

Example:

1. 3^{rd} root of -8 is ^{3}√-8

And ^{3}√-8 = (-8)^{1/3} = (-2^{3})^{1/3} = (-2)^{ 3×1/3} = 2^{3×1/3} = 2

But, 2^{nd} root of -4 does not exist, since 2 is an even integer and if the exponent is even the base can’t be negative.

Note:

nth root of a positive number “a” is not defined in real numbers if n is an even integer, but

nth root of the positive number “a” exists in the set of complex numbers even if n is an even integer.

The Table summarizes all the laws of exponents.

Logarithmic form.

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