**How to express a given surd to simplest form :**

Here we are going to see how to express a given surd to its simplest form.

To reduce the given surd to its simplest form, we have to follow the steps given below.

**Step 1 :**

Find factors of the number inside the given radical.

**Step 2 :**

Based on the order of the radical, we have to group them as pairs and factor out.

**Step 3 :**

In case we already have terms outside the radical, we have to multiply them by the factor taken out and multiply the remaining numbers inside the radical.

Let us look into some example problems based on the above concept.

**Example 1 :**

Express the following surd in its simplest form.

^{3}√32

**Solution :**

^{3}√32 = ^{3}√(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)

Order of the given radical is 3.

So, we have to factor out one term for every three same terms.

= 2 ^{3}√(2 ⋅ 2)

= 2 ^{3}√4

Hence the simplified form of the given radical term ^{3}√32 is 2 ^{3}√4.

**Example 2 :**

Express the following surd in its simplest form.

√63

**Solution :**

√63 = √(3 ⋅ 3 ⋅ 7)

Order of the given radical is 2.

So, we have to factor out one term for every two same terms.

= 3 √7

Hence the simplified form of the given radical term √63 is 3√7.

**Example 3 :**

Express the following surd in its simplest form.

√243

**Solution :**

√243 = √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3)

Order of the given radical is 2.

So, we have to factor out one term for every two same terms.

= (3 ⋅ 3) √3

= 9 √3

Hence the simplified form of the given radical term √243 is 9√3.

**Example 4 :**

Express the following surd in its simplest form.

^{3}√256

**Solution :**

^{3}√256 = ^{3}√(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)

Order of the given radical is 3.

So, we have to factor out one term for every three same terms.

= 2 ⋅ 2 ⋅ 2 ⋅ 2

= 16

Hence the simplified form of the given radical term ^{3}√256 is 16.

**Example 5 :**

Express the following surd in its simplest form.

^{4}√80

**Solution :**

^{4}√80 = ^{4}√(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5)

Order of the given radical is 4.

So, we have to factor out one term for every four same terms.

= 2 ^{4}√5

Hence the simplified form of the given radical term ^{4}√80 is 2 ^{4}√5.

- Rationalization of surds
- Comparison of surds
- Operations with radicals
- Ascending and descending order of surds
- Simplifying radical expression
- Exponents and powers

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