# nth Roots

**Real numbers** are the numbers that include both **rational and irrational numbers**. Rational numbers such as integers (-2, 0, 1), fractions(1/2, 2.5), and irrational numbers such as √3, π(22/7), etc., are all real numbers. The definition of the nth root of a real number can be stated as:

For any two real numbers a and b, and any positive integer n, if a^{n }= b, then a is an nth root of b.

**For Example:**

3^{4 }= 81

Where,

a = 3, b = 81, and n = 4That means that 3 is the fourth root of 81.

**How to write nth Roots?**

indicates an nth root.

**For Example:**

Where,

144 = Radicand,

√ = Radical Sign, and

n = IndexThat means that we are taking the nth root of 144. One can take any value of n, like n = 2, 3, 5, etc. We will be taking that root of 100.

**Principal Root**

Some numbers have more than one real nth root. For example, 36 has two roots, one is +6 and another is -6. In this case, the non-negative root is called the **principal root**.

**When no index is given the radical sign indicates the principal root.** For example, if we take then we are talking about its principal root which is 6.

### Examples

**Example 1: If n = 5 and b = 32 then a = ?**

**Solution:**

Given:a^{5 }= 32=>

2 . 2 . 2 . 2 . 2 = 32

Since, 2 is the fifth root of 32

Therefore, a = 2

**Example 2: n = 6 and b = 4096 then a = ?**

**Solution:**

Given:a^{6 }= 4096=>

To find the value of a, we will check which integer value is the sixth root of 4096

2 . 2 . 2 . 2 . 2 . 2 = 64

3 . 3 . 3 . 3 . 3 . 3 = 729

4 . 4 . 4 . 4 . 4 . 4 = 4096

Since, 4 is the sixth root of 4096

Therefore, a = 4

**Example 3: If a = 18, b = 2, n = 2, and** **then find the value of c?**

**Solution:**

Given:

SInce the index value is same then the radicand is multiplied

Since, 6 is the square root of 36 (6

^{2}= 36)Therefore, c = 6

**Example 4: If a = 5, b = 15, n = 3 and****then find the value of c?**

**Solution:**

Given:

SInce the index value is same then the radicand are divided

**Example 5: Solve the following:**

**Solution:**

Given:

1 . 1 . 1 = 1, 1 is the cube root of 1

2 . 2 . 2 = 8, 2 is the cube root of 8

3 . 3 . 3 = 27, 3 is the cube root of 27

4 . 4 . 4 = 64, 4 is the cube root of 64

c = 1 + 2 + 3 + 4

c = 10

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