Algebra Galore 1

(4x + 3)^2/3 =9

Raise each side to the power 3/2

4x + 3 = 9^3/2

4x + 3 = 27

4x = 24

x=6

\qquad \qquad { (4x+3) }^{ \frac { 2 }{ 3 } }=9\\ Take\quad power\quad \frac { 3 }{ 2 } \quad on\quad both\quad sides.\\ we\quad have\\ \qquad (4x+3)={ 9 }^{ \frac { 3 }{ 2 } }\quad =>\quad 4x+3=\sqrt { 729 } \\ =>\quad 4x+3=27\quad =>\quad 4x=24\\ =>\quad x=\frac { 24 }{ 4 } \quad or\quad \boxed { x=6 }

Given:

$(4x+3)^{\frac{2}{3}}=9$

Cubing both sides:

$(4x+3)^{2}=9^{3}$

$16x^{2}+24x+9=729$

$16x^{2}+24x-720=0$

$8(2x^{2}+3x-90)=0$

Dividing both sides by $8$ :

$2x^{2}+3x-90=0$

$2x^{2}-12x+15x-90=0$

$2x(x-6)+15(x-6)=0$

$(x-6)(2x+15)=0$

So, the values of $x=6$ and $-7.5$

But, $x$ cannot be negative.

Thus, the answer is: $x=\boxed{6}$