What Is The Simplest Form Of $\sqrt[3]{x^{10}}$?A. $3 \sqrt[3]{x}$B. $x \sqrt[3]{x}$C. $x^3 \sqrt[3]{x}$D. $3 X \sqrt[3]{x}$

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Understanding the Problem

When dealing with radicals, it's essential to simplify them to their most basic form. In this case, we're given the expression $\sqrt[3]{x^{10}}$, and we need to determine its simplest form. To do this, we'll apply the properties of radicals and exponents.

Properties of Radicals and Exponents

Before we dive into the solution, let's review some key properties of radicals and exponents:

• Product of Powers Property: When multiplying two powers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Property: When raising a power to another power, we multiply their exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Root of a Power Property: When taking the root of a power, we divide the exponent by the index of the root. For example, $\sqrt[n]{a^m} = a^{\frac{m}{n}}$.

Simplifying the Expression

Now that we've reviewed the properties of radicals and exponents, let's apply them to simplify the expression $\sqrt[3]{x^{10}}$.

We can start by using the Power of a Power Property to rewrite $x^{10}$ as $(x^3)^3 \cdot x$. This gives us:

$$\sqrt[3]{x^{10}} = \sqrt[3]{(x^3)^3 \cdot x}$$

Next, we can use the Product of Powers Property to separate the terms inside the radical:

$$\sqrt[3]{(x^3)^3 \cdot x} = \sqrt[3]{(x^3)^3} \cdot \sqrt[3]{x}$$

Now, we can use the Root of a Power Property to simplify the first term:

$$\sqrt[3]{(x^3)^3} = x^3$$

So, we have:

$$\sqrt[3]{x^{10}} = x^3 \cdot \sqrt[3]{x}$$

Final Answer

Therefore, the simplest form of $\sqrt[3]{x^{10}}$ is:

$$x^3 \sqrt[3]{x}$$

This is the correct answer, which corresponds to option C.

Conclusion

In this article, we've demonstrated how to simplify the expression $\sqrt[3]{x^{10}}$ using the properties of radicals and exponents. By applying these properties, we were able to rewrite the expression in its simplest form, which is $x^3 \sqrt[3]{x}$. This result is essential in various mathematical applications, and it's crucial to understand how to simplify radicals to their most basic form.

Frequently Asked Questions

• What is the simplest form of $\sqrt[3]{x^5}$? The simplest form of $\sqrt[3]{x^5}$ is $x \sqrt[3]{x}$.
• How do I simplify a radical expression? To simplify a radical expression, you can use the properties of radicals and exponents, such as the product of powers property, power of a power property, and root of a power property.
• What is the difference between a radical and an exponent? A radical is a mathematical operation that involves taking the root of a number, while an exponent is a mathematical operation that involves raising a number to a power.

Additional Resources

• Radical Expressions: A radical expression is a mathematical expression that involves a root, such as $\sqrt{x}$ or $\sqrt[3]{x}$.
• Exponent Properties: Exponent properties are mathematical rules that govern how to manipulate exponents, such as the product of powers property and the power of a power property.
• Simplifying Radicals: Simplifying radicals involves rewriting them in their most basic form using the properties of radicals and exponents.

Final Thoughts

In conclusion, simplifying radical expressions is a crucial skill in mathematics, and it's essential to understand how to apply the properties of radicals and exponents to rewrite expressions in their simplest form. By mastering this skill, you'll be able to tackle a wide range of mathematical problems and applications with confidence.

Q&A: Simplifying Radical Expressions

In this article, we'll address some of the most common questions related to simplifying radical expressions. Whether you're a student, a teacher, or simply someone looking to brush up on their math skills, this article is for you.

Q: What is the simplest form of $\sqrt[3]{x^5}$?

A: The simplest form of $\sqrt[3]{x^5}$ is $x \sqrt[3]{x}$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can use the properties of radicals and exponents, such as the product of powers property, power of a power property, and root of a power property.

Q: What is the difference between a radical and an exponent?

A: A radical is a mathematical operation that involves taking the root of a number, while an exponent is a mathematical operation that involves raising a number to a power.

Q: Can I simplify a radical expression with a negative exponent?

A: Yes, you can simplify a radical expression with a negative exponent by using the property of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$.

Q: How do I simplify a radical expression with a variable in the radicand?

A: To simplify a radical expression with a variable in the radicand, you can use the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: Can I simplify a radical expression with a coefficient?

A: Yes, you can simplify a radical expression with a coefficient by factoring out the coefficient and simplifying the resulting expression.

Q: How do I simplify a radical expression with multiple terms?

A: To simplify a radical expression with multiple terms, you can use the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: Can I simplify a radical expression with a rational exponent?

A: Yes, you can simplify a radical expression with a rational exponent by using the property of rational exponents, which states that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Q: How do I simplify a radical expression with a complex number?

A: To simplify a radical expression with a complex number, you can use the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: Can I simplify a radical expression with a trigonometric function?

A: Yes, you can simplify a radical expression with a trigonometric function by using the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: How do I simplify a radical expression with a logarithmic function?

A: To simplify a radical expression with a logarithmic function, you can use the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: Can I simplify a radical expression with a hyperbolic function?

A: Yes, you can simplify a radical expression with a hyperbolic function by using the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: How do I simplify a radical expression with a mixed number?

A: To simplify a radical expression with a mixed number, you can use the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: Can I simplify a radical expression with a decimal number?

A: Yes, you can simplify a radical expression with a decimal number by using the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: How do I simplify a radical expression with a fraction?

A: To simplify a radical expression with a fraction, you can use the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: Can I simplify a radical expression with a negative number?

A: Yes, you can simplify a radical expression with a negative number by using the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: How do I simplify a radical expression with a zero exponent?

A: To simplify a radical expression with a zero exponent, you can use the property of zero exponents, which states that $a^0 = 1$.

Q: Can I simplify a radical expression with a negative exponent and a variable?

A: Yes, you can simplify a radical expression with a negative exponent and a variable by using the property of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$.

Q: How do I simplify a radical expression with a rational exponent and a variable?

A: To simplify a radical expression with a rational exponent and a variable, you can use the property of rational exponents, which states that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Q: Can I simplify a radical expression with a complex number and a variable?

A: Yes, you can simplify a radical expression with a complex number and a variable by using the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: How do I simplify a radical expression with a trigonometric function and a variable?

A: To simplify a radical expression with a trigonometric function and a variable, you can use the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: Can I simplify a radical expression with a logarithmic function and a variable?

A: Yes, you can simplify a radical expression with a logarithmic function and a variable by using the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: How do I simplify a radical expression with a hyperbolic function and a variable?

A: To simplify a radical expression with a hyperbolic function and a variable, you can use the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: Can I simplify a radical expression with a mixed number and a variable?

A: Yes, you can simplify a radical expression with a mixed number and a variable by using the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: How do I simplify a radical expression with a decimal number and a variable?

A: To simplify a radical expression with a decimal number and a variable, you can use the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: Can I simplify a radical expression with a fraction and a variable?

A: Yes, you can simplify a radical expression with a fraction and a variable by using the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: How do I simplify a radical expression with a negative number and a variable?

A: To simplify a radical expression with a negative number and a variable, you can use the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: Can I simplify a radical expression with a zero exponent and a variable?

A: Yes, you can simplify a radical expression with a zero exponent and a variable by using the property of zero exponents, which states that $a^0 = 1$.

Q: How do I simplify a radical expression with a negative exponent and a variable?

A: To simplify a radical expression with a negative exponent and a variable, you can use the property of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$.

Q: Can I simplify a radical expression with a rational exponent and a variable?

A: Yes, you can simplify a radical expression with a rational exponent and a variable by using the property of rational exponents, which states that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Q: How do I simplify a radical expression with a complex number and a variable?

A: To simplify a radical expression with a complex number and a variable, you can use the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: Can I simplify a radical expression with a trigonometric function and a variable?

A: Yes, you can simplify a radical expression with a trigonometric function and a variable by using the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: How do I simplify a radical expression with a logarithmic function and a variable?

A: To simplify a radical expression with a logarithmic function and a variable, you can use the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: Can I simplify a radical expression with a hyperbolic function and a variable?

A: Yes, you can simplify a radical expression with a hyperbolic function and a variable by using the properties of radicals and exponents, such as the product of powers property and the power of a power property.

Q: How do I simplify a radical expression with a mixed number and a variable?

A: To simplify a radical expression with a mixed number and a variable, you can use the properties of radicals and exponents, such as the product of powers property and the power of