How to solve a math problem

Solve and check for extraneous solutions.
1. (4x+3)^2/3 = (16x+44)^1/3

Please explain how to do this, I am really confused and want to understand it.

(4x+3)^(2/3) = (16x+44)^(1/3)

Expand the left side:
(4x+3)^(1/3) * (4x+3)^(1/3) = (16x+44)^(1/3)

Combine the left side:
[(4x+3)(4x+3)]^(1/3) = (16x+44)^(1/3)

Cube both sides:
{[(4x+3)(4x+3)]^(1/3)}^3 = [(16x+44)^(1/3)]^3
(4x+3)(4x+3) = 16x+44

Do FOIL on the left side:
16x^2+12x+12x+9 = 16x+44
16x^2+24x+9 = 16x+44

Subtract 16x from both sides:
16x^2+24x+9-16x = 16x+44-16x
16x^2+8x+9 = 44

Subtract 44 from both sides:
16x^2+8x+9-44 = 44-44
16x^2+8x-35 = 0

Factor the trinomial on the left side:
(4x-5)(4x+7) = 0

Set each factor equal to 0 and solve:
Either 4x-5=0 or 4x+7=0.
Therefore x=5/4 or x=-7/4.

Check:

[4(5/4)+3]^(2/3) = [16(5/4)+44]^(1/3)
(5+3)^(2/3) = (20+44)^(1/3)
8^(2/3) = 64^(1/3)
Cube root 8^2 = Cube root 64
Cube root 64 = Cube root 64
Correct

[4(-7/4)+3]^(2/3) = [16(-7/4)+44]^(1/3)
(-7+3)^(2/3) = (-28+44)^(1/3)
(-4)^(2/3) = 16^(1/3)
Cube root (-4)^2 = Cube root 16
Cube root 16 = Cube root 16
Correct

Answer: x = -7/4, 5/4

First note that if a = b, then a^n = b^n

So from (4x+3)^2/3 = (16x+44)^1/3
Take the cube of both sides
(4x+3)^2 = (16x+44)
Expand out the LHS
16x^2 + 24x + 9 = 16x + 44
Bring all terms to the LHS
16x^2 + 8x - 35 = 0
And solve the quadratic equation using the formula x = (-b +/- Sqrt(b^2 - 4ac)/2a,
where the quadratic is of the form ax^2 + bx + c = 0
with a = 16, b = 8 and c = -35
x = (-8 +/- Sqrt(8^2 - 4*16*(-35))) / (2*16)
= (-8 +/- Sqrt(64 + 2240)) / 32
= (-8 +/- Sqrt(2304))/32
= (-8 +/- 48) /32
So the two solutions are
x = (-8 - 48) /32= -56/32 = 1.75
x = (-8 + 48) /32 = 40/32 = 1.25

cube (3rd power) both sides and get
(4x+3)² = (16x+44)¹
16x² + 24x + 9 = 16x + 44
16x² + 8x – 35 = 0 ∙ ∙ ∙ ∙ ∙ ∙ ∙ now factor
16x² + 28x – 20x – 35 = 0
4x(4x + 7) – 5(4x + 7) = 0
(4x – 5)(4x + 7) = 0
x = 5/4 or x = -7/4

check them, both solutions are good.

(4x+3)^2/3 = (16x+44)^1/3
(4x+3)^2 = 16x+44
16x^2+24x+9 = 16x+44
16x^2+8x-35 = 0
(4x+7)(4x-5)=0
x = -7/4 or x = 5/4
Check x = 5/4
8^(2/3) = 4
64^(1/3) = 4 So True
Check x = -7/4
-4^(2/3) = 2.52
16^(1/3) = 2.52 so True
x = -7/4 or 5/4