20. \( -3 x\left(x^{\frac{1}{4}}+1\right)^{2} \)

It looks like you have the expression: \[ -3x\left(x^{\frac{1}{4}} + 1\right)^{2} \] Let's expand and simplify this expression step by step. ### Step 1: Expand the Squared Term First, expand \(\left(x^{\frac{1}{4}} + 1\right)^{2}\) using the binomial expansion formula \((a + b)^2 = a^2 + 2ab + b^2\): \[ \left(x^{\frac{1}{4}} + 1\right)^{2} = \left(x^{\frac{1}{4}}\right)^2 + 2\left(x^{\frac{1}{4}} \cdot 1\right) + 1^2 = x^{\frac{1}{2}} + 2x^{\frac{1}{4}} + 1 \] ### Step 2: Distribute \(-3x\) Across the Expanded Terms Now, multiply \(-3x\) with each term inside the parentheses: \[ -3x \cdot \left(x^{\frac{1}{2}} + 2x^{\frac{1}{4}} + 1\right) = -3x \cdot x^{\frac{1}{2}} - 3x \cdot 2x^{\frac{1}{4}} - 3x \cdot 1 \] ### Step 3: Simplify Each Term 1. **First Term:** \[ -3x \cdot x^{\frac{1}{2}} = -3x^{1 + \frac{1}{2}} = -3x^{\frac{3}{2}} \] 2. **Second Term:** \[ -3x \cdot 2x^{\frac{1}{4}} = -6x^{1 + \frac{1}{4}} = -6x^{\frac{5}{4}} \] 3. **Third Term:** \[ -3x \cdot 1 = -3x \] ### Step 4: Combine All Terms Putting it all together: \[ -3x\left(x^{\frac{1}{4}} + 1\right)^{2} = -3x^{\frac{3}{2}} - 6x^{\frac{5}{4}} - 3x \] ### Final Expanded Form \[ -3x^{\frac{3}{2}} - 6x^{\frac{5}{4}} - 3x \] If you prefer to write the exponents with radicals: \[ -3x^{\frac{3}{2}} = -3x \sqrt{x}, \quad -6x^{\frac{5}{4}} = -6x \sqrt[4]{x}, \quad \text{and} \quad -3x \text{ remains the same}. \] So, another way to write the expanded expression is: \[ -3x\sqrt{x} - 6x\sqrt[4]{x} - 3x \] Feel free to let me know if you need further assistance with this expression!