Simplify: $64^{\frac{1}{3}}$

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Introduction

When dealing with exponents and roots, it's essential to understand the properties and rules that govern them. In this article, we will focus on simplifying the expression $64^{\frac{1}{3}}$. This involves understanding the concept of cube roots and how to apply them to simplify expressions.

Understanding Cube Roots

A cube root of a number is a value that, when multiplied by itself twice, gives the original number. In mathematical terms, if $x = \sqrt[3]{y}$, then $x^3 = y$. The cube root of a number can be denoted as $\sqrt[3]{y}$ or $y^{\frac{1}{3}}$.

Simplifying the Expression

To simplify the expression $64^{\frac{1}{3}}$, we need to find the cube root of 64. Since 64 is a perfect cube (i.e., it can be expressed as a power of 4), we can rewrite it as $4^3$. Therefore, the cube root of 64 is equal to the cube root of $4^3$.

Applying the Property of Cube Roots

Using the property of cube roots, we can rewrite the expression as follows:

6413=643=43364^{\frac{1}{3}} = \sqrt[3]{64} = \sqrt[3]{4^3}

Simplifying Further

Since the cube root of a power is equal to the power of the cube root, we can simplify the expression further:

433=(43)13\sqrt[3]{4^3} = (4^3)^{\frac{1}{3}}

Applying the Power of a Power Rule

Using the power of a power rule, we can rewrite the expression as follows:

(43)13=43×13(4^3)^{\frac{1}{3}} = 4^{3 \times \frac{1}{3}}

Simplifying the Exponent

Simplifying the exponent, we get:

43×13=414^{3 \times \frac{1}{3}} = 4^1

Final Answer

Therefore, the simplified expression is:

6413=464^{\frac{1}{3}} = 4

Conclusion

In this article, we simplified the expression $64^{\frac{1}{3}}$ by applying the properties of cube roots and exponents. We showed that the cube root of 64 is equal to the cube root of $4^3$, and then simplified the expression further using the power of a power rule. The final answer is $4$.

Frequently Asked Questions

Q: What is the cube root of 64?

A: The cube root of 64 is equal to the cube root of $4^3$, which is equal to 4.

Q: How do you simplify an expression with a cube root?

A: To simplify an expression with a cube root, you can apply the property of cube roots, which states that the cube root of a power is equal to the power of the cube root.

Q: What is the power of a power rule?

A: The power of a power rule states that for any numbers a and b and any integer n, $(ab)n = a^{b \times n}$.

Additional Resources

For more information on exponents and roots, check out the following resources:

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Introduction

In our previous article, we simplified the expression $64^{\frac{1}{3}}$ by applying the properties of cube roots and exponents. In this article, we will answer some frequently asked questions related to the simplification of this expression.

Q&A

Q: What is the cube root of 64?

A: The cube root of 64 is equal to the cube root of $4^3$, which is equal to 4.

Q: How do you simplify an expression with a cube root?

A: To simplify an expression with a cube root, you can apply the property of cube roots, which states that the cube root of a power is equal to the power of the cube root.

Q: What is the power of a power rule?

A: The power of a power rule states that for any numbers a and b and any integer n, $(ab)n = a^{b \times n}$.

Q: Can you explain the concept of cube roots in more detail?

A: A cube root of a number is a value that, when multiplied by itself twice, gives the original number. In mathematical terms, if $x = \sqrt[3]{y}$, then $x^3 = y$. The cube root of a number can be denoted as $\sqrt[3]{y}$ or $y^{\frac{1}{3}}$.

Q: How do you simplify an expression with a cube root and a power?

A: To simplify an expression with a cube root and a power, you can apply the property of cube roots and the power of a power rule. For example, to simplify $64^{\frac{1}{3}}$, you can rewrite it as $\sqrt[3]{64} = \sqrt[3]{4^3}$, and then simplify further using the power of a power rule.

Q: Can you provide more examples of simplifying expressions with cube roots?

A: Yes, here are a few more examples:

  • 2723=2723=(272)13=272327^{\frac{2}{3}} = \sqrt[3]{27^2} = (27^2)^{\frac{1}{3}} = 27^{\frac{2}{3}}

  • 832=833=(83)13=818^{\frac{3}{2}} = \sqrt[3]{8^3} = (8^3)^{\frac{1}{3}} = 8^1

  • 1643=1643=(164)13=164316^{\frac{4}{3}} = \sqrt[3]{16^4} = (16^4)^{\frac{1}{3}} = 16^{\frac{4}{3}}

Q: How do you know when to use the power of a power rule?

A: You should use the power of a power rule when you have an expression with a cube root and a power. For example, if you have $64^{\frac{1}{3}}$, you can rewrite it as $\sqrt[3]{64} = \sqrt[3]{4^3}$, and then simplify further using the power of a power rule.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $64^{\frac{1}{3}}$. We provided explanations and examples to help clarify the concepts of cube roots and the power of a power rule. We hope this article has been helpful in understanding these important mathematical concepts.

Additional Resources

For more information on exponents and roots, check out the following resources:

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