How to simplify radical expressions with variables and exponents :
To simplify radical terms or radical expressions first we have to find the factors.
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 y³x 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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