20. $$ x^{\frac{2}{3}}-5 x^{\frac{1}{3}}+6=0 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
x^{\frac{2}{3}} - 5x^{\frac{1}{3}} + 6 = 0
$$

**Step 1: Substitution**

Let’s make a substitution to simplify the equation. Let:

$$
y = x^{\frac{1}{3}}
$$

Then, the equation becomes:

$$
y^2 - 5y + 6 = 0
$$

**Step 2: Solve the Quadratic Equation**

Now, solve for $$ y $$:

$$
y^2 - 5y + 6 = 0
$$

Factor the quadratic:

$$
(y - 2)(y - 3) = 0
$$

So, the solutions for $$ y $$ are:

$$
y = 2 \quad \text{or} \quad y = 3
$$

**Step 3: Back-Substitute to Find $$ x $$**

Recall that $$ y = x^{\frac{1}{3}} $$. Therefore:

1. If $$ y = 2 $$:
   $$
   x^{\frac{1}{3}} = 2 \implies x = 2^3 = 8
   $$

2. If $$ y = 3 $$:
   $$
   x^{\frac{1}{3}} = 3 \implies x = 3^3 = 27
   $$

**Final Solutions:**

$$
x = 8 \quad \text{and} \quad x = 27
$$
The solutions are $$ x = 8 $$ and $$ x = 27 $$.

20. \( x^{\frac{2}{3}}-5 x^{\frac{1}{3}}+6=0 \)