Question

Solve the following mathematical equation: 

$(3x +1)^{\frac{3}{2}} - 24 = 60$

Answer 1

To find the value of $x$, we will assume:

$3x + 1 = u$

Now, substituting $3x + 1 = u$ into the given equation, we have:

$u^{\frac{3}{2}} - 24 = 40$

$\Rightarrow u^{\frac{3}{2}} = 40 + 24$

$\Rightarrow u^{\frac{3}{2}} = 64$

Squaring both sides, we have:

$\Rightarrow \left( u^{\frac{3}{2}} \right)^2 = (64)^2$

$\Rightarrow u^3 = (64)^2$

$\Rightarrow u = \left( 64 \right)^{\frac{2}{3}} \, \, \, \, \left[ \because a^{n} = b \,\, \therefore a = b^{\frac{1}{n}} \right]$

$\Rightarrow u = (4^3)^{\frac{2}{3}}$

$\Rightarrow u = 4^2$

$\Rightarrow u = 16$

Now, undo substitution $u = 16$ into $3x + 1 = u$ , we have:

$3x + 1 = 16$

$\Rightarrow 3x = 16 - 1$

$\Rightarrow x = \dfrac{15}{3}$

$\Rightarrow x = 5$

 

Hence, the solution of the given equation is $x = \color{blue}{\boxed{ 5 }}$