**Evaluate integers raised to rational exponents :**

Here we are going to learn about how to evaluate integers raised to rational exponents.

**Step 1 :**

Try to split the number which is in base as much as possible.

**Step 2 :**

Whenever we have power raised to another power, then we can multiply both the powers. If it is possible we can simplify.

Let us see some examples based on the above concept.

**Example 1 :**

Simplify 9^(1/2)

**Solution :**

**We can write 9 as the multiple of 3,**

**9 = 3 x 3 = 3**²

9^(1/2) = (3²)^(1/2)

= 3 ^(2 x 1/2)

= 3 ^1

= 3

Hence 3 is the answer.

**Example 2 :**

Simplify 343^(-4/3)

**Solution :**

**We can write 343 as the multiple of 7,**

**343 = 7**** x 7**** x 7**** = 7**³

343^(-4/3) = (**7**³)^(-4/3)

= **7^3 **x (-4/3)

= 7^-4

= 1/7^4

= 1/2401

Hence 1/2401 is the answer.

**Example 3 :**

Simplify 36^(3/2)

**Solution :**

**We can express 36 as the multiple of 6,**

**36 = 6**** x 6**** = 6**²

36^(3/2) = (6²)^(3/2)

= 6**^2 **x (3/2)

= 6^3

= 6 x 6 x 6

= 216

Hence 216 is the answer.

**Example 4 :**

Simplify (81 m⁶)^(1/2)

**Solution :**

**We can express 81 as the multiple of 9,**

**81 = 9**** x 9**** = 9**²

(81 m⁶)^(1/2) = (**9**²m⁶)^(1/2)

By distributing the power, we get

= (9²)^(1/2) (m⁶)^(1/2)

= 9 m^(6 x 1/2)

= 9 m^3

Hence 9 m^3 is the answer.

**Example 5 :**

Simplify (64 n^12)^(1/6)

**Solution :**

**We can express 64 as the multiple of 2,**

** 64 = 2**** x 2 ****x 2 ****x 2 ****x 2 ****x 2**** = 2**⁶

(64 n^12)^(1/6) = (**2**⁶ n^12)^(1/6)

By distributing the power, we get

= (**2**⁶)^(1/6) (n^12)^(1/6)

= 2 n^(12 x 1/6)

= 2 n^2

Hence 2 n^2 is the answer.

**Example 6 :**

Simplify (x^6)^(1/2)

**Solution :**

= x^(6(1/2))

= x^3

Hence x^3 is the answer.

**Example 7 :**

Simplify (9n^4)^(1/2)

**Solution :**

**We can express 9 as the multiple of 3,**

** 9 = 3**** x 3 = 3**²

(9n^4)^(1/2) = (3²n^4)^(1/2)

By distributing the power, we get

= (3²)^(1/2) (n^4)^(1/2)

= 3^(2 x 1/2) (n^(4 x 1/2)

= 3 n^2

Hence 3 n^2 is the answer.

**Example 8 :**

Simplify (27)^(1/6)

**Solution :**

**We can express 27 as the multiple of 3,**

** 27 = 3**** x 3 x 3 = 3**³

(27)^(1/6) = ( 3³)^(1/6)

By multiplying the power raised to another power, we get

= 3^3 x (1/6)

= 3^(1/2)

= √3

Hence √3 is the answer.

**Example 9 :**

Simplify (100)^(6/4)

**Solution :**

**We can express 100 as the multiple of 10,**

** 100 = 10**** x 10 = 10**²

(100)^(6/4) = (**10**²)^(6/4)

= 1**0^2 x **(6/4)

= 1**0^2 x **(3/2)

= 1**0^3**

= 10 x 10 x 10 = 1000

Hence the answer is 1000.

**Example 10 :**

Simplify (64a^6)^(1/6)

**Solution :**

**We can express 64 as the multiple 2,**

**64 = 2 x 2 x 2 x ****2 x 2 x 2**

** = 2^6**

**(64a^6)^(1/6) = (2^6a^6)^(1/6)**

** = (2a)^6^(1/6)**

** = 2a**

**Hence the answer is 2a.**

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