5) $$ 1-4(3 x-3)^{\frac{3}{2}}=-107 $$

Question

UpStudy AI · Accepted Answer

Let's solve the equation step by step:

$$
1 - 4(3x - 3)^{\frac{3}{2}} = -107
$$

**Step 1: Isolate the Term with the Exponent**

Start by moving the constant term to the other side:

$$
-4(3x - 3)^{\frac{3}{2}} = -107 - 1
$$

$$
-4(3x - 3)^{\frac{3}{2}} = -108
$$

**Step 2: Eliminate the Coefficient**

Divide both sides by $$-4$$ to isolate the exponential term:

$$
(3x - 3)^{\frac{3}{2}} = \frac{-108}{-4}
$$

$$
(3x - 3)^{\frac{3}{2}} = 27
$$

**Step 3: Remove the Exponent**

To eliminate the exponent $$\frac{3}{2}$$, raise both sides to the power of $$\frac{2}{3}$$:

$$
\left( (3x - 3)^{\frac{3}{2}} \right)^{\frac{2}{3}} = 27^{\frac{2}{3}}
$$

$$
3x - 3 = 27^{\frac{2}{3}}
$$

**Step 4: Simplify the Right Side**

Recall that $$27^{\frac{1}{3}} = 3$$, so:

$$
27^{\frac{2}{3}} = \left(27^{\frac{1}{3}}\right)^2 = 3^2 = 9
$$

$$
3x - 3 = 9
$$

**Step 5: Solve for $$x$$**

Add 3 to both sides:

$$
3x = 12
$$

Divide by 3:

$$
x = 4
$$

**Solution:** $$x = 4$$
$$ x = 4 $$

5) \( 1-4(3 x-3)^{\frac{3}{2}}=-107 \)