Simplify: $\sqrt[4]{256 X^4 Y^8}$A. $2|x| Y^2$ B. $4|x| Y^2$ C. $4 X^2 Y^4$ D. $16\left|x^2\right| Y^4$

Understanding the Problem

The given problem involves simplifying a radical expression, which is a mathematical expression that contains a root or a power. In this case, we are dealing with a fourth root, denoted by the symbol $\sqrt[4]{}$. The expression inside the radical is $256 x^4 y^8$. Our goal is to simplify this expression and find the most simplified form.

Breaking Down the Expression

To simplify the expression, we need to break it down into its prime factors. The number $256$ can be expressed as $2^8$, and the variables $x^4$ and $y^8$ can be expressed as $x^4$ and $y^8$, respectively. Therefore, the expression can be rewritten as:

$$\sqrt[4]{2^8 x^4 y^8}$$

Applying the Properties of Radicals

Now that we have broken down the expression, we can apply the properties of radicals to simplify it. One of the properties of radicals is that we can take the root of a product as the product of the roots. In other words, $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. We can apply this property to the expression as follows:

$$\sqrt[4]{2^8 x^4 y^8} = \sqrt[4]{2^8} \cdot \sqrt[4]{x^4} \cdot \sqrt[4]{y^8}$$

Simplifying the Radicals

Now that we have applied the property of radicals, we can simplify each radical individually. The fourth root of $2^8$ is $2^2$, the fourth root of $x^4$ is $x$, and the fourth root of $y^8$ is $y^2$. Therefore, the expression can be simplified as follows:

$$\sqrt[4]{2^8 x^4 y^8} = 2^2 \cdot x \cdot y^2$$

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it. The expression $2^2 \cdot x \cdot y^2$ can be evaluated as follows:

$$2^2 \cdot x \cdot y^2 = 4 \cdot x \cdot y^2$$

Conclusion

In conclusion, the simplified form of the expression $\sqrt[4]{256 x^4 y^8}$ is $4|x| y^2$. This is because the absolute value of $x$ is included in the simplified expression.

Answer

The correct answer is B. $4|x| y^2$.

Explanation

The correct answer is $4|x| y^2$ because the absolute value of $x$ is included in the simplified expression. The absolute value of $x$ is denoted by $|x|$, and it is included in the simplified expression because the fourth root of $x^4$ is $x$. Therefore, the correct answer is $4|x| y^2$.

Final Answer

Q: What is the simplified form of the expression $\sqrt[4]{256 x^4 y^8}$?

A: The simplified form of the expression $\sqrt[4]{256 x^4 y^8}$ is $4|x| y^2$. This is because the absolute value of $x$ is included in the simplified expression.

Q: Why is the absolute value of $x$ included in the simplified expression?

A: The absolute value of $x$ is included in the simplified expression because the fourth root of $x^4$ is $x$. When we take the fourth root of $x^4$, we get $x$, which includes the absolute value of $x$.

Q: What is the property of radicals that we used to simplify the expression?

A: The property of radicals that we used to simplify the expression is that we can take the root of a product as the product of the roots. In other words, $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$.

Q: How did we simplify the radicals in the expression?

A: We simplified the radicals in the expression by taking the fourth root of each factor individually. The fourth root of $2^8$ is $2^2$, the fourth root of $x^4$ is $x$, and the fourth root of $y^8$ is $y^2$.

Q: What is the final answer to the problem?

A: The final answer to the problem is B. $4|x| y^2$.

Q: Why is the final answer $4|x| y^2$?

A: The final answer is $4|x| y^2$ because the absolute value of $x$ is included in the simplified expression. The absolute value of $x$ is denoted by $|x|$, and it is included in the simplified expression because the fourth root of $x^4$ is $x$.

Q: What is the importance of simplifying radicals?

A: Simplifying radicals is important because it helps us to evaluate expressions more easily. By simplifying radicals, we can reduce the complexity of the expression and make it easier to work with.

Q: How can we apply the property of radicals to simplify other expressions?

A: We can apply the property of radicals to simplify other expressions by taking the root of each factor individually. For example, if we have the expression $\sqrt[4]{a^4 b^8 c^12}$, we can simplify it by taking the fourth root of each factor individually: $\sqrt[4]{a^4} \cdot \sqrt[4]{b^8} \cdot \sqrt[4]{c^{12}}$.

Q: What are some common mistakes to avoid when simplifying radicals?

A: Some common mistakes to avoid when simplifying radicals include:

• Not taking the root of each factor individually
• Not simplifying the radicals correctly
• Not including the absolute value of variables in the simplified expression

Q: How can we check our work when simplifying radicals?

A: We can check our work when simplifying radicals by plugging the simplified expression back into the original expression and evaluating it. If the simplified expression is correct, it should evaluate to the same value as the original expression.