To solve the given expression $\left(\sqrt[4]{x} + \sqrt[4]{x^2}\right)^2$, let's go step by step.

Step 1: Understand the components of the expression.

• $\sqrt[4]{x}$ means the fourth root of $x$, which is $x^{1/4}$.
• $\sqrt[4]{x^2}$ means the fourth root of $x^2$, which can be written as $x^{2/4}$ or $x^{1/2}$.

Thus, the expression can be rewritten as: $\left(x^{1/4} + x^{1/2}\right)^2$

Step 2: Expand the expression.

To expand $\left(x^{1/4} + x^{1/2}\right)^2$, we use the formula for the square of a binomial: $(a + b)^2 = a^2 + 2ab + b^2$ Here, $a = x^{1/4}$ and $b = x^{1/2}$. Applying the formula, we get: $\left(x^{1/4} + x^{1/2}\right)^2 = \left(x^{1/4}\right)^2 + 2\cdot x^{1/4} \cdot x^{1/2} + \left(x^{1/2}\right)^2$

Step 3: Simplify each term.

1. $\left(x^{1/4}\right)^2 = x^{2/4} = x^{1/2}$
2. $2 \cdot x^{1/4} \cdot x^{1/2}$: To simplify this, add the exponents since the bases are the same: $x^{1/4} \cdot x^{1/2} = x^{1/4 + 1/2} = x^{1/4 + 2/4} = x^{3/4}$ Thus, $2 \cdot x^{3/4}$
3. $\left(x^{1/2}\right)^2 = x^{1}$, because $x^{1/2 \cdot 2} = x^{1}$.

Step 4: Combine all terms.

$x^{1/2} + 2x^{3/4} + x^{1}$ This is the expanded and simplified form of the given expression.

Final Answer:

The expression $\left(\sqrt[4]{x} + \sqrt[4]{x^2}\right)^2$ simplifies to: $x^{1/2} + 2x^{3/4} + x$

Would you like further details or have any questions on this?

Related Questions:

1. How would the expression change if the original power was a cube root instead of a fourth root?
2. What happens if $x$ is negative? Does the expression still hold?
3. Can this expression be factored back into its original form? If so, how?
4. What is the derivative of the simplified expression with respect to $x$?
5. How does this expression behave as $x$ approaches infinity?

Tip:

When simplifying expressions involving roots and powers, always remember to combine like terms by adding or subtracting exponents accordingly.