**Simplifying square root expressions with variables :**

Here we are going to see how to simplify square root expressions with variables.

The following examples will illustrate the method of simplifying square root expression with variables.

**Example 1 :**

Simplify (10 + √3) (2 + √5)

**Solution :**

(10 + √3) (2 + √5)

To find the product of the given expressions, we have to use the distributive property.

= (10 + √3) (2 + √5)

= 10 (2) + 10 √5 + √3 (2) + √3 (√5)

= 20 + 10 √5 + 2√3 + √(3 ⋅ 5)

= 20 + 10 √5 + 2√3 + √15

No two terms are not having same radicand, so we cannot combine the terms.

Hence the answer is 20 + 10 √5 + 2√3 + √15.

**Example 2 :**

Simplify (√5 + √3)^{2}

**Solution :**

(√5 + √3)^{2}

The given expression exactly matches with the algebraic identity (a + b)^{2}

(a + b)^{2 } = a^{2} + 2ab + b^{2}

a = √5 and b = √3

(√5 + √3)^{2 } = (√5)^{2 }+ 2 (√5)(√3) + (√3)^{2}

= 5^{ }+ 2 √(5 ⋅ 3) + 3

= 5 + 3^{ }+ 2 √15

= 8 + 2 √15

Hence the answer is 8 + 2 √15.

**Example 3 :**

Simplify (√13 - √2)(√13 + √2)

**Solution :**

(√13 - √2)(√13 + √2)

The given expression exactly matches with the algebraic identity (a + b)(a - b)

(a + b)(a - b) = a^{2} - b^{2}

(√13 - √2)(√13 + √2) = (√13)^{2} - (√2)^{2}

= 13 - 12

= 1

**Example 4 :**

Simplify (8 + √3)(8 - √3)

**Solution :**

(8 + √3)(8 - √3)

The given expression exactly matches with the algebraic identity (a + b)(a - b)

(a + b)(a - b) = a^{2} - b^{2}

(8 + √3)(8 - √3) = (8)^{2} - (√3)^{2}

= 64 - 3

= 61

**Example 5 :**

Simplify (5 + √3)(8 - 2√5)

**Solution :**

(5 + √3)(8 - 2√5)

To find the product of the given expressions, we have to use the distributive property.

= (5 + √3)(8 - 2√5)

= 5 (8) + 5 (-2√5) + √3 (8) + √3 (-2√5)

= 40 - 10 √5 + 8√3 - 2√(3 ⋅ 5)

= 40 - 10 √5 + 8√3 - 2√15

No two terms are not having same radicand, so we cannot combine the terms.

Hence the answer is 40 - 10 √5 + 8√3 - 2√15.

- 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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