radical solutions

Question

Solve the equation. Check for extraneous solutions.

1)3√(x+3)-12=6
2)4(x-4)^1/4-3=5
3)(x)^1/3-5=0
4)x^3/4-8=19
5)2x^1/3=-6

Answer 1

1)3√(x+3)-12 = 6

Apply addition property, add 12 to each sides.

3√(x+3)-12+12 = 6+12

3√(x+3) = 18

Apply divisible property,divide by 3 each sides.

3√(x+3)/3 = 18/3

√(x+3) = 6

(x+3) = 6^2

(x+3) = 6×6

x+3 = 36

Apply subtraction property,subtract 3 from each sides.

x+3-3 = 36-3

⇒x = 33

check:

Substitute the value of x in eqation.

3√(33+3)-12 = 6

3√36-12 = 6

3×6-12 = 6

18-12 = 6

6 = 6

2)4(x-4)^1/4-3 = 5

Apply addition property, add 3 to each sides.

4(x-4)^1/4-3+3 = 5+3

4(x-4)^1/4= 8

Apply division property, divide by 4 to each sides.

4(x-4)^1/4/4= 8/4

(x-4)^1/4 = 2

x-4 = 2⁴

x-4 = 2×2×2×2

x-4 = 16

Apply addition property,add 4 to each sides.

x-4+4 = 16+4

⇒x = 20

check:

Substitute the value of x in eqation.

4(20-4)^1/4-3 = 5

4(16)^1/4-3 = 5

Aplly addition property,add 3 to each sides

4(16)^1/4-3+3 = 5+3

4(16)^1/4 = 8.

Apply divisible property, divide by 4 to each sides.

4(16)^1/4/4 = 8/4

(16)^1/4 = 2

16 = 2⁴

16 = 2×2×2×2

16 = 16

3)(x)^1/3-5=0

Apply addition property, add 5 to each sides.

(x)^1/3-5+5=0+5

(x)^1/3=5

x = 5³

x = 5×5×5

⇒x = 125

Check:

Substitute the x value in equation

(125)^1/3-5 = 0

Apply addition property,add5 to each sides.

(125)^1/3-5+5 = 0+5

(125)^1/3 = 5

125 = 5³  

125 = 5×5×5

125 = 125

Answer 2

4).

The equation is (x)^3/4 - 8 = 19.

Apply addition property, add 8 to each side.

(x)^3/4 - 8 + 8 = 19 + 8

(x)^3/4 = 27

(x)^3/4 = 3³

x = 3^{3 * 4/3}

x = 3⁴

x = 3 × 3 × 3 x 3

⇒x = 81.

The solution is x = 81.

Check:

Substitute the x value in the original equation

(81)^3/4-8 = 19

Apply addition property, add 8 to each side.

(81)^3/4- 8 + 8 = 19 + 8

(81)^3/4 = 27

81 = (3³)^3/4

81 = 3^3*(4/3)

81 = 3⁴

81 = 3 × 3 x 3 × 3

81 = 81.

The above statement is true.

So, x = 81 is the solution of (x)^3/4 - 8 = 19.

Answer 3

5).

The equation is 2x^1/3  = - 6.

Divide each side by 2.
x^1/3  = - 6/2

x^1/3  = - 3

x = (- 3)³

x = (- 3) × (- 3) × (- 3)

⇒x = - 27.

The solution is x = - 27.

Check:

Substitute the x value in the original equation

2(- 27)^1/3 = - 6

Divide each side by 2.

(- 27)^1/3 = - 3

- 27 = (- 3)³

- 27 = (- 3) × (- 3) x (- 3)

- 27 = - 27.

The above statement is true.

So, x = - 27 is the solution of 2x^1/3 = - 6.