**Properties of exponents :**

The exponent of a number says how many times to use the number in a multiplication.

for example 5³ = 5 x 5 x 5

In words 5³ could be called as 5 to the power 3 or 5 cube.

**Product of Powers Property :**

When we have to simplify two or more the terms which are multiplying with same base, we have to put the same base and add the powers.

**Quotient of Powers Property :**

Whenever we have two terms which are diving with the same base, we have to put only one base and we have to subtract the powers.

**Power of a Power Property :**

**Whenever we have power to the power, we have to multiply both powers. **

**Zero Exponent Rule :**

**Anything to the power zero is 1.**

**Power of a Product Property :**

If we have same power for 2 or more terms which are multiplying or dividing, we have to apply the powers for every terms.

**Example 1 :**

Simplify 2m^{2} ⋅ 2m^{3}

**Solution :**

= 2m^{2} ⋅ 2m^{3}

= 4m^{(2+3)}

= 4m^{5}

Since we have same base for both terms, we put only one base and add the powers.

Hence the answer is 4m^{5}

**Example 2 :**

Simplify m^{4} ⋅ 2m^{-3}

**Solution :**

= m^{4} ⋅ 2m^{-3}

= 2m^{(4 - 3)}

= 2m^{1}

Since we have same base for both terms, we put only one base and combining the powers.

Hence the answer is 2m.

**Example 3 :**

Simplify (4a^{3})^{2}

**Solution :**

= (4a^{3})^{2}

Since we have power 2 for both the terms 4 and a^{3}, we need to distribute the power for both terms

= 4^{2}(a^{3})^{2}

= 16a^{6}

Hence the answer is 16a^{6}

**Example 4 :**

Simplify (x^{3})^{0}

**Solution :**

= (x^{3})^{0}

Since we have power zero, the answer must be 1.

**Example 5 :**

Simplify 4 a^{3 }b^{2 }⋅ 3 a^{4 }b^{3}

**Solution :**

= 4 a^{3 }b^{2 }⋅ 3 a^{4 }b^{3}

= 12 a^{(3+4)}b^{(2+3)}

= 12 a^{7}b^{5}

Hence the answer is 12 a^{7}b^{5}

**Example 6 :**

Simplify r^{2}/2r^{3}

**Solution :**

= r^{2}/2r^{3}

= (1/2) r^{(2-3)}

= (1/2) r^{-1}

= (1/2r)

Hence the answer is 1/2r.

**Example 7 :**

Simplify 4x^{0}y^{2}z^{3}/4x

**Solution :**

= 4x^{0}y^{2}z^{3}/4x

The value of anything to the power zero is 1. So, x^{0} is 1.

= y^{2}z^{3}/x

**Example 8 :**

Simplify (2xy^{4})^{4}(xy)^{5}

**Solution :**

= 2^{4}x^{4 }(y^{4})^{4 }x^{5 }y^{5}

= 16 x^{4 }y^{16 }x^{5 }y^{5}

= 16 x^{(4 + 5) }y^{(16 + 5)}

= 16 x^{9 }y^{21}

Hence the answer is 16 x^{9 }y^{21}

**Example 9 :**

Simplify x^{-4}/x^{-9}

**Solution :**

= x^{-4}/x^{-9}

= x^{-4}x^{9}

= x^{(-4 + 9)}

= x^{5}

Hence the answer is x^{5}

**Example 10 :**

Simplify x^{-1}/4x^{4}

**Solution :**

= x^{-1}/4x^{4}

= (1/4) x^{ (-1 + 4)}

= (1/4) x^{ 3}

= 1/4 x^{3}

Hence the answer is 1/4 x^{3}

After having gone through the stuff given above, we hope that the students would have understood, "Properties of exponents".

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