Determine the sum of the roots of the following equation:

2
-1
1
-2

Ask by Washington Weston. in Pakistan

Mar 22,2025

Question

Determine the sum of the roots of the following equation: $$ 3^{\sqrt[3]{x^{2}}}=\frac{1}{3} \cdot 3^{\sqrt[3]{x}+1} $$ 2 -1 1 -2

UpStudy AI · Accepted Answer

Let $$ 3^{\sqrt[3]{x^2}} = \frac{1}{3} \cdot 3^{\sqrt[3]{x}+1} $$.

1. Rewrite the right-hand side by noting that $$\frac{1}{3} = 3^{-1}$$:
   $$
   \frac{1}{3} \cdot 3^{\sqrt[3]{x}+1} = 3^{-1} \cdot 3^{\sqrt[3]{x}+1} = 3^{\sqrt[3]{x}+1-1} = 3^{\sqrt[3]{x}}
   $$

2. With the equation simplified, we have:
   $$
   3^{\sqrt[3]{x^2}} = 3^{\sqrt[3]{x}}
   $$
   Since the bases are equal, the exponents must be equal:
   $$
   \sqrt[3]{x^2} = \sqrt[3]{x}
   $$

3. Cube both sides to eliminate the cube roots:
   $$
   (\sqrt[3]{x^2})^3 = (\sqrt[3]{x})^3 \quad \Longrightarrow \quad x^2 = x
   $$

4. Rearrange the equation:
   $$
   x^2 - x = 0
   $$
   Factor:
   $$
   x(x-1) = 0
   $$

5. Set each factor equal to zero:
   $$
   x=0 \quad \text{or} \quad x=1
   $$

6. The sum of the roots is:
   $$
   0 + 1 = 1
   $$

The sum of the roots is $$ 1 $$.
The sum of the roots is 1.

Determine the sum of the roots of the following equation:

2
-1
1
-2

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Determine the sum of the roots of the following equation: 2 -1 1 -2