**How to simplify radical expressions with variables and exponents :**

To simplify radical terms or radical expressions first we have to find the factors.

- According to the index of the given radical, we have to take one common term from the radical.
- For example if we have square root, then we have to take one common term for every two same terms.
- If we have cube root then we have to take one common term for every three same terms.
- In case we have algebraic expression in the radical sign, we have to find factors using algebraic identities.

Here we can find some frequently used tables while finding the square root of a number and also you can find frequently used algebraic formulas.

Let us see some example problems based on the above concept.

**Example 1 :**

Find the square root of 196 a⁶ b⁸ c¹⁰

**Solution :**

= √(196 a⁶ b⁸ c¹⁰ )

= √(14 x 14 a³ x a³ x b⁴ x b⁴ x c⁵ x c⁵)

= 14 |a³b⁴ c⁵|

**Example 2 :**

Find the square root of 289 (a - b)⁴ (b - c)⁶

**Solution :**

= √289 (a - b)⁴ (b - c)⁶

= √(17 x 17 (a - b)² (a - b)² (b - c)³(b - c)³

= 17 |(a - b)² (b - c)³|

**Example 3 :**

Find the square root of (x + 11)² - 44 x

**Solution :**

(a + b)² = a² + 2 a b + b²

= √(x² + 2 x (11) + 11² - 44 x)

= √(x² + 22 x + 121 - 44 x)

= √(x² - 22 x + 121 )

= √(x - 11)²

= |x - 11|

**Example 4 :**

Find the square root of (x - y)² + 4 x y

**Solution :**

(a - b)² = a² - 2 a b + b²

= √[(x - y)² + 4 x y]

= √[x² - 2 x y + y² + 4 x y]

= √(x² + 2 x y + y²)

= √((x + y)²

= (x + y)

**Example 5 :**

Find the square root of 121 x⁸ y⁶ ÷ 81 x⁴ y⁸

**Solution:**

= √(121 x⁸ y⁶/ 81 x⁴ y⁸)

= √(11 **x** 11 **x** x⁴ **x** x⁴ **x **y³**x **y³/ 9** x** x²** x** x² **x **y⁴ **x **y⁴)

= (11/9) ( x⁴ ** **y³/ x²** **y⁴)

= (11/9) ( x² /y)

**Example 6 :**

Find the square root of [64 (a + b)⁴(x - y)⁸(b - c)⁶]/[25 (x+ y)⁴ (a - b)⁶(b + c)¹⁰]

**Solution:**

= √[64 (a + b)⁴(x - y)⁸(b - c)⁶]/[25 (x+ y)⁴ (a - b)⁶(b + c)¹⁰]

= (8/5)|[(a + b)² (x - y)⁴(b - c)³/(x+ y) ² (a - b)³(b + c)⁵]|

**Example 7 :**

Find the square root of 16 x² - 24 x + 9

**Solution :**

= √(16 x² - 24 x + 9)

= √(4 x)² - 2 (4x) (3) + 3²

= √(4 x - 3)²

= |(4 x - 3)|

**Example 8 :**

Find the square root of (x² - 25)(x² + 8 x + 15)(x² - 2 x - 15)

**Solution :**

= √(x² - 25)(x² + 8 x + 15)(x² - 2 x - 15)

= √(x + 5)(x - 5) (x + 3) (x + 5) (x - 5)(x + 3)

= |(x - 5) (x + 5) (x + 3)|

**Example 9 :**

Find the square root of 4x² + 9y² + 25z² - 12xy + 30 yz - 20 zx How to simplify radical expressions with variables and exponents

**Solution :**

= √(4x² + 9y² + 25z² - 12xy + 30 yz - 20 zx)

= √(2x)²+(-3y)²+(-5z)²+2(2x)(-3y)+2(-3y)(-5z)+2(2x)(-5z)

= √(2x - 3y - 5z)²

= |(2x - 3y - 5z)|

**Example 10 :**

Find the square root of x⁴ + (1/x⁴) + 2

**Solution :**

(a + b)² = a² + 2 a b + b²

= √(x²)² + (1/x²)² + 2 x² (1/x²)]

= √(x² + (1/x²))²

= |(x² + (1/x²))|

After having gone through the stuff given above, we hope that the students would have understood "How to simplify radical expressions with variables and exponents".

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