Applying the Rule for Radicals

Oct 13, 2025 by recoveryshake.com 136 views

Understanding the Problem

When dealing with exponents and roots, it's essential to understand the properties and rules that govern them. In this problem, we're given the expression $$24^{\frac{1}{3}}$$ and asked to find an equivalent expression from the given options. To solve this, we need to apply the rules of exponents and roots, specifically the rule for fractional exponents.

Fractional Exponents

A fractional exponent is a shorthand way of expressing a root. For example, $$a^{\frac{1}{n}}$$ is equivalent to $$\sqrt[n]{a}$$. This rule allows us to rewrite the given expression in a more familiar form.

Rewriting the Expression

Using the rule for fractional exponents, we can rewrite $$24^{\frac{1}{3}}$$ as $$\sqrt[3]{24}$$. This is because the exponent $$\frac{1}{3}$$ indicates that we need to take the cube root of the number inside the radical.

Simplifying the Radical

Now that we have $$\sqrt[3]{24}$$, we can simplify the radical by factoring the number inside it. We can write $$24$$ as $$2^3 \cdot 3$$, which means that $$\sqrt[3]{24}$$ is equivalent to $$\sqrt[3]{2^3 \cdot 3}$$.

Applying the Rule for Radicals

When we have a product inside a radical, we can take the cube root of each factor separately. This means that $$\sqrt[3]{2^3 \cdot 3}$$ is equivalent to $$\sqrt[3]{2^3} \cdot \sqrt[3]{3}$$.

Simplifying the Expression

Now that we have $$\sqrt[3]{2^3} \cdot \sqrt[3]{3}$$, we can simplify the expression further. The cube root of $$2^3$$ is simply $$2$$, so we're left with $$2 \cdot \sqrt[3]{3}$$.

Comparing with the Options

Now that we have $$2 \cdot \sqrt[3]{3}$$, we can compare it with the given options. The only option that matches this expression is $$2 \sqrt[3]{3}$$.

Conclusion

In conclusion, the expression $$24^{\frac{1}{3}}$$ is equivalent to $$2 \sqrt[3]{3}$$. This is because we can rewrite the expression using the rule for fractional exponents, simplify the radical, and apply the rule for radicals to get the final result.

Final Answer

The final answer is: $\boxed{2 \sqrt[3]{3}}$

Frequently Asked Questions

Q: What is the rule for fractional exponents?

A: The rule for fractional exponents is that $$a^{\frac{1}{n}}$$ is equivalent to $$\sqrt[n]{a}$$.

Q: How do I simplify a radical?

A: To simplify a radical, you can factor the number inside it and take the root of each factor separately.

Q: What is the rule for radicals?

A: The rule for radicals is that when you have a product inside a radical, you can take the root of each factor separately.

Q: How do I compare an expression with the given options?

A: To compare an expression with the given options, you need to simplify the expression and then match it with the options.

Additional Resources

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

A: An exponent is a number that is raised to a power, while a root is the inverse operation of an exponent. For example, $$2^3$$ is an exponent, while $$\sqrt[3]{8}$$ is a root.

Q: How do I simplify a radical?

A: To simplify a radical, you can factor the number inside it and take the root of each factor separately. For example, $$\sqrt{12}$$ can be simplified as $$\sqrt{4 \cdot 3}$$, which is equal to $$2\sqrt{3}$$.

Q: What is the rule for radicals?

A: The rule for radicals is that when you have a product inside a radical, you can take the root of each factor separately. For example, $$\sqrt{12}$$ can be simplified as $$\sqrt{4 \cdot 3}$$, which is equal to $$2\sqrt{3}$$.

Q: How do I compare an expression with the given options?

A: To compare an expression with the given options, you need to simplify the expression and then match it with the options. For example, if you have the expression $$2^3$$ and the options $$2^2$$, $$2^4$$, and $$2^5$$, you can simplify the expression to $$8$$ and then match it with the options.

Q: What is the difference between a square root and a cube root?

A: A square root is the inverse operation of squaring a number, while a cube root is the inverse operation of cubing a number. For example, $$\sqrt{16}$$ is equal to $$4$$, while $$\sqrt[3]{27}$$ is equal to $$3$$.

Q: How do I simplify an expression with multiple roots?

A: To simplify an expression with multiple roots, you can use the rule for radicals, which states that when you have a product inside a radical, you can take the root of each factor separately. For example, $$\sqrt{12}$$ can be simplified as $$\sqrt{4 \cdot 3}$$, which is equal to $$2\sqrt{3}$$.

Q: What is the rule for exponents?

A: The rule for exponents states that when you have a product of numbers raised to a power, you can add the exponents. For example, $$2^3 \cdot 2^4$$ is equal to $$2^{3+4}$$, which is equal to $$2^7$$.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can use the rule for exponents, which states that when you have a product of numbers raised to a power, you can add the exponents. For example, $$2^3 \cdot 2^4$$ is equal to $$2^{3+4}$$, which is equal to $$2^7$$.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent is a number that is raised to a power, while a negative exponent is the inverse operation of a positive exponent. For example, $$2^3$$ is a positive exponent, while $$2^{-3}$$ is a negative exponent.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can use the rule for negative exponents, which states that $$a^{-n}$$ is equal to $$\frac{1}{a^n}$$. For example, $$2^{-3}$$ is equal to $$\frac{1}{2^3}$$, which is equal to $$\frac{1}{8}$$.

Q: What is the rule for fractional exponents?

A: The rule for fractional exponents states that $$a^{\frac{1}{n}}$$ is equal to $$\sqrt[n]{a}$$. For example, $$2^{\frac{1}{3}}$$ is equal to $$\sqrt[3]{2}$$.

Q: How do I simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, you can use the rule for fractional exponents, which states that $$a^{\frac{1}{n}}$$ is equal to $$\sqrt[n]{a}$$. For example, $$2^{\frac{1}{3}}$$ is equal to $$\sqrt[3]{2}$$.

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

A: A rational exponent is a number that can be expressed as a fraction, while an irrational exponent is a number that cannot be expressed as a fraction. For example, $$2^{\frac{1}{3}}$$ is a rational exponent, while $$\sqrt{2}$$ is an irrational exponent.

Q: How do I simplify an expression with a rational exponent?

A: To simplify an expression with a rational exponent, you can use the rule for rational exponents, which states that $$a^{\frac{m}{n}}$$ is equal to $$(\sqrt[n]{a})^m$$. For example, $$2^{\frac{1}{3}}$$ is equal to $$(\sqrt[3]{2})^1$$, which is equal to $$\sqrt[3]{2}$$.

Q: What is the rule for multiplying exponents?

A: The rule for multiplying exponents states that when you have a product of numbers raised to a power, you can add the exponents. For example, $$2^3 \cdot 2^4$$ is equal to $$2^{3+4}$$, which is equal to $$2^7$$.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can use the rule for multiplying exponents, which states that when you have a product of numbers raised to a power, you can add the exponents. For example, $$2^3 \cdot 2^4$$ is equal to $$2^{3+4}$$, which is equal to $$2^7$$.

Q: What is the rule for dividing exponents?

A: The rule for dividing exponents states that when you have a quotient of numbers raised to a power, you can subtract the exponents. For example, $$2^3 \div 2^4$$ is equal to $$2^{3-4}$$, which is equal to $$2^{-1}$$.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can use the rule for dividing exponents, which states that when you have a quotient of numbers raised to a power, you can subtract the exponents. For example, $$2^3 \div 2^4$$ is equal to $$2^{3-4}$$, which is equal to $$2^{-1}$$.

Q: What is the rule for exponents with the same base?

A: The rule for exponents with the same base states that when you have a product or quotient of numbers raised to a power with the same base, you can add or subtract the exponents. For example, $$2^3 \cdot 2^4$$ is equal to $$2^{3+4}$$, which is equal to $$2^7$$, and $$2^3 \div 2^4$$ is equal to $$2^{3-4}$$, which is equal to $$2^{-1}$$.

Q: How do I simplify an expression with multiple exponents with the same base?

A: To simplify an expression with multiple exponents with the same base, you can use the rule for exponents with the same base, which states that when you have a product or quotient of numbers raised to a power with the same base, you can add or subtract the exponents. For example, $$2^3 \cdot 2^4$$ is equal to $$2^{3+4}$$, which is equal to $$2^7$$, and $$2^3 \div 2^4$$ is equal to $$2^{3-4}$$, which is equal to $$2^{-1}$$.

Q: What is the rule for exponents with different bases?

A: The rule for exponents with different bases states that when you have a product or quotient of numbers raised to a power with different bases, you cannot add or subtract the exponents. For example, $$2^3 \cdot 3^4$$ is not equal to $$2^{3+4}$$, which is equal to $$2^7$$, and $$2^3 \div 3^4$$ is not equal to $$2^{3-4}$$, which is equal to $$2^{-1}$$.

Q: How do I simplify an expression with multiple exponents with different bases?

A: To simplify an expression with multiple exponents with different bases, you cannot use the rule for exponents with the same base. Instead, you need to simplify each exponent separately and then multiply or divide the results. For example, $$2^3 \cdot 3^4$$ is equal to $$8 \cdot 81$$, which is equal to $$648$$, and $$2^3 \div 3^4$$ is equal to $$8 \div 81$$, which is equal to $$\frac{8}{81}$$.