Once you have $3x^{3/2}=24$, what simple operation can you use to get $x^{3/2}$ by itself on one side?

Think of $x^{3/2}$ as $(\sqrt{x})^3$. After you set $(\sqrt{x})^3$ equal to a number, what root can you take to simplify it?

After you find $\sqrt{x}$, what operation will give you $x$ from its square root?

In Desmos, type each expression on its own line:

• 3*(2)^(3/2)
• 3*(4)^(3/2)
• 3*(8)^(3/2)
• 3*(16)^(3/2)

Look at the numerical values Desmos displays for each line; the correct choice is the one whose value equals $24$, since that is when $g(x)=24$.

We are told that $g(x)=3x^{3/2}$ and asked for which $x$ we have $g(x)=24$.

So set up the equation:

Divide both sides of the equation by $3$ to isolate the term with $x$:

Now we need to solve $x^{3/2} = 8$.

The exponent $\tfrac{3}{2}$ means "cube, then take a square root" or "take a square root, then cube". A helpful way to see this is:

So the equation $x^{3/2} = 8$ becomes

Let $y = \sqrt{x}$. Then we have

Taking the cube root of both sides gives $y = 2$, so $\sqrt{x} = 2$.

From $\sqrt{x}=2$, square both sides to solve for $x$:

Check: $g(4) = 3\cdot 4^{3/2} = 3\cdot (\sqrt{4})^3 = 3\cdot 2^3 = 3\cdot 8 = 24$, so it works.

Thus, the value of $x$ is $4$, which corresponds to answer choice B) 4.