Solve the equation. $$ (x+2)^{2 / 5}-5(x+2)^{1 / 5}+6=0 $$ $$ x=1 $$ Simplify your answer. Type an integer or a sim

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
(x+2)^{2/5} - 5(x+2)^{1/5} + 6 = 0
$$

**Step 1:** Let’s make a substitution to simplify the equation.

Let $$ y = (x + 2)^{1/5} $$. Then, $$ y^2 = (x + 2)^{2/5} $$.

**Step 2:** Substitute $$ y $$ into the equation:

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

**Step 3:** Factor the quadratic equation:

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

**Step 4:** Solve for $$ y $$:

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

**Step 5:** Recall that $$ y = (x + 2)^{1/5} $$. Solve for $$ x $$ in each case.

1. When $$ y = 2 $$:

$$
(x + 2)^{1/5} = 2 \
x + 2 = 2^5 \
x + 2 = 32 \
x = 30
$$

2. When $$ y = 3 $$:

$$
(x + 2)^{1/5} = 3 \
x + 2 = 3^5 \
x + 2 = 243 \
x = 241
$$

**Solution:**

$$
x = 30 \quad \text{and} \quad x = 241
$$

**Final Answer:**

**30, 241**
The solutions are $$ x = 30 $$ and $$ x = 241 $$.

Solve the equation. \[ (x+2)^{2 / 5}-5(x+2)^{1 / 5}+6=0 \] \( x=1 \) Simplify your answer. Type an integer or a sim