**Multiplication with rational exponents :**

When we have two two or more terms with same bases and different rational exponents then we have to write one base instead of those bases and add the powers.

When we have a term raised to the power by another power then we have to multiply those powers.

**Example 1 :**

Evaluate the following

**Solution :**

**Here we have same power for the numerator and the denominator. **

** = [(42/6)^(1/3)]^1/2**

** = [7^(1/3)]^1/2**

**Since we have power raised to another power, we have to multiply these two powers.**

** = 7^(1/6)**

**Hence, 7^(1/6) is the answer.**

**Example 2 :**

**Solution :**

** = 1/x^4**

**Hence, ****1/x^4**** is the answer.**

**Example 3 :**

Evaluate the following

**Solution :**

Instead of square root we can use 1/2 as power and cube root we can use 1/3 as power.

25 x^16 = (5 x^8)^2 [25 = 5 **x** 5]

= [x^(4/3) **x** (x)^(5/2)]/[5 x^8)^2]^(1/2)

= x^[(4/3) + (5/2)]/(5 x^8)

= x^[(8 + 15)/6]/(5 x^8)

= x^(23/6)/(5 x^8)

= (1/5) x^[(23/6) - 8]

= (1/5) x^(-25/6)

= 1/5 x^(25/6)

Hence 1/5 x^(25/6) is the answer.

**Example 4 :**

Evaluate the following

12^(1/8) **x** 12^(5/6)

**Solution :**

= 12^(1/8) **x** 12^(5/6)

= 12^[(1/8) + (5/6)]

To add the fractions 1/8 and 5/6 we have take L.C.M for the denominators.

L.C.M of (8, 6) = 24

To make 8 as 24 we have to multiply

= 12^[(3 + 20)/24]

= 12^(23/24)

Hence 12^(23/24) is the answer.

**Example 5 :**

Evaluate the following

[5^(1/3) **x** x^(1/4)]^3

**Solution :**

= [5^(1/3) **x** x^(1/4)]^3

Now we have to distribute power 3 for both 5^(1/3) and x^(1/4). So it becomes

= 5 **x** x^(3/4)

= 5 x^(3/4)

Hence 5 x^(3/4) is the answer.

**Example 6 :**

Evaluate the following

√(8 x^3) **x** √(18 x)

**Solution :**

= √(8 x^3) **x** √(18 x)

√8 = √(2 **x **2 **x** 2 x^3) = 2x√(2 x)

√18 x = √(3 **x** 3 **x **2 **x** x) = 3√(2 x)

√(8 x^3) **x** √(18 x) = [2x√(2 x)] [3√(2 x)]

= 6 x (2 x)

= 12 x^2

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