EVALUATE RADICAL EXPRESSIONS

To evaluate a radical expression, replace the variables in the expression by the given values of the variables and simplify order of operations.

After having replaced the variables by the given values, write the number inside the radical sign as a product of its factors.

Take one number out of the radical for every two same numbers multiplied inside the radical sign, if the radical is a square root.

Take one number out of the radical for every three same numbers multiplied inside the radical sign, if the radical is a cube root.

Examples :

√4  =  √(2  2)  =  2

√16  =  √(2  2  2  2)  =  2  2  =  2

3√27  =  3√(3  3  3)  =  3

3√125  =  3√(5  5  5)  =  5

Evaluate the following expressions using the given values of the variables.

Example 1 :

√x + √y for x = 4 and y = 9

Solution :

= √x + √y

Substitute x = 4 and y = 9.

= √4 + √9

= 2 + 3

= 5

Example 2 :

2√(185 - x) for x = 169

Solution :

= 2√(185 - x)

Substitute x = 169.

= 2√(185 - 169)

= 2√16

= 2√(4 ⋅ 4)

= 2(4)

= 8

Example 3 :

5√x3 for x = 3

Solution :

= 5√x3

Substitute x = 3.

5√33

= 5√(3 ⋅ 3 ⋅ 3)

= 5(3√(3)

= 15√3

Example 4 :

5x + 1 for x = 1

Solution :

= 5x + 1

Substitute x = 1.

= 5√1 + 1

= 5√1 + 1

= 5(1) + 1

= 5 + 1

= 6

Example 5 :

√(x/49) for x = 25 

Solution :

= √(x/49)

Substitute x = 25.

√(25/49)

√25/√49

√(5 ⋅ 5)/√(7 ⋅ 7)

= 5/7

Example 6 :

√(x2 + y2) for x = 3 and y = 4

Solution :

= √(x2 + y2)

Substitute x = 3 and y = 4.

= √(32 + 42)

= √(9 + 16)

= √25

= √(5 ⋅ 5)

= 5

Example 7 :

3√x - 3√y for x = 27 and y = 8

Solution :

3√x - 3√y

Substitute x = 27 and y = 8.

= 3√27 - 3√8

= 3 - 2

= 1

Example 8 :

√12w + √27w for w = 3

Solution :

= √(12w) + √(27w)

= √(2  2 ⋅  w) + √(3  3 ⋅  w)

= 2√3w + 3√3w

= 5√3w

Substitute w = 3.

= 5√(3 ⋅ 3)

= 5(3)

= 15

Example 9 :

3√8x3y6 + √9x2y4 for x = 1 and y = 2

Solution :

3√8x3y6 + √9x2y4

3√(2 ⋅  2  x  x  x  y y y2) + √(3  3  x  x  y2  y2)

= 2xy2 + 3xy2

= 5xy2

Substitute x = 1 and y = 2.

= 5(1)(22)

= 5(1)(4)

= 20

Example 10 :

√4p2q4 - 3√125p3q6 for p = -2 and q = -3

Solution :

= √4p2q4 - 3√125p3q6

= √(2 ⋅  p  p  q q2) - √(5  5  5  p  p  p ⋅ q2  q q2)

= 2pq2 - 5pq2

= -3pq2

Substitute p = -2 and q = -3.

= -3(-2)(-3)2

= -3(-2)(9)

= 54 

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