**Power rule of exponents :**

Here we are going to see what are power rules of exponents.

**Case 1 :**

When we have a power for any number or variable raised to another power, then we have to multiply both the powers.

**Case 2 :**

In case we have two monomials which are multiplying or dividing then we have to consider the power as common for both terms.So we have to distribute the power.

**Case 3 :**

In case we have negative power then we have to take the reciprocal of the given term and change negative power as positive.

3^{-1} = 1/3 (Reciprocal of 3 is 1/3)

(2/5)^{-7} = (5/2)^{7} (Reciprocal of 2/5 is 5/2)

**Case 4 :**

Anything to the power zero is 1.

**Example 1 :**

Simplify (6 t^{6})^{2}

**Solution :**

** = **(6 t^{6})^{2}

**Here the power 2 is common for both 6 and t^6.Since it is multiplying, we have to distribute the power for both terms. **

** = 6 ^{2} (t^{6})^{2}**

** = 36 t ^{(6 x 2)}**

** = 36 t ^{12}**

**Example 2 :**

Simplify (2/u)^{5}

**Solution :**

= (2/u)^{5}

**Here the power 5 is common for both numerator and denominator.**** **

** ** = (2

** ** = (32/u

**Example 3 :**

Simplify (5c^{7})^{2}

**Solution :**

= (5c^{7})^{2}

**Here the power 2 is common for both 5 and c ^{7}**

** = 5 ^{2} (c^{7})^{2}**

** = (5 x 5)[ c ^{(7 x 2)} ]**

** = 25 c ^{14}**

**Example 4 :**

Simplify (10 t^{3})^{2}

**Solution :**

= (10 t^{3})^{2}

**Here the power 2 is common for both 10 and t ^{3}**

** = **10^{2} (t^{3})^{2}

** = (10 x 10)[ t ^{(3 x 2)} ]**

** = 100 t ^{6}**

**Example 5 :**

Simplify the expression given below.

4³· 5³

**Solution : **

**Using power of a product property, we can simplify the given expression.**

4³· 5³ ** = (4 **· 5)³

4³· 5³ ** = 20**³

4³· 5³** = 8000**

**Example 6 :**

Simplify the expression given below.

128⁰· 128

**Solution : **

**Using zero exponent property, we can simplify the given expression.**

128⁰· 128 = 1 · 128

128⁰· 128 = 128

**Example 7 :**

Simplify the expression given below.

5⁻³

**Solution :**

**Using negative exponent property, we can simplify the given expression.**

** **5⁻³ **= 1/5**³

**= 1/125**

**Example 8 :**

Simplify the expression given below.

(a^{1/2})^{3/2}

**Solution :**

**When we have power raised to another power, we have to multiply both powers.**

(a^{1/2})^{3/2 } = a^{(1/2)(3/2)}

= a^{(3/4)}

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