Simplifying the Fractional Exponent

Oct 13, 2025 by windailys.com 13 views

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Introduction

In mathematics, the cube root of a number is a value that, when multiplied by itself twice, gives the original number. The cube root of a number can be denoted by the symbol $\sqrt[3]{x}$ , where $x$ is the number inside the radical sign. In this article, we will focus on rewriting the cube root of 128 in its simplest form.

Understanding Cube Roots

A cube root is a type of radical expression that involves the extraction of a cube root of a number. The cube root of a number $x$ is denoted by $\sqrt[3]{x}$ and is defined as the number that, when multiplied by itself twice, gives the original number $x$ . For example, the cube root of 27 is 3, because $3 \times 3 \times 3 = 27$ .

Breaking Down 128

To rewrite $\sqrt[3]{128}$ in its simplest form, we need to break down 128 into its prime factors. The prime factorization of 128 is $2^7$ . This means that 128 can be expressed as the product of 2 raised to the power of 7.

Simplifying the Cube Root

Now that we have broken down 128 into its prime factors, we can simplify the cube root of 128. We know that the cube root of a number can be rewritten as the product of the cube roots of its prime factors. Therefore, we can rewrite $\sqrt[3]{128}$ as $\sqrt[3]{2^7}$ .

Applying the Power Rule

The power rule of exponents states that for any positive integer $n$ , $(a^m)^n = a^{mn}$ . We can apply this rule to simplify the cube root of 2 raised to the power of 7. This means that we can rewrite $\sqrt[3]{2^7}$ as $2^{7/3}$ .

Simplifying the Fractional Exponent

The fractional exponent $7/3$ can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 1. Therefore, the fractional exponent $7/3$ remains the same.

Final Answer

The final answer to the problem of rewriting $\sqrt[3]{128}$ in its simplest form is $2^{7/3}$ . This is the simplest form of the cube root of 128, and it cannot be simplified further.

Conclusion

In this article, we have learned how to rewrite the cube root of 128 in its simplest form. We started by breaking down 128 into its prime factors, and then we applied the power rule of exponents to simplify the cube root of 2 raised to the power of 7. Finally, we simplified the fractional exponent to obtain the final answer of $2^{7/3}$ . This is an important concept in mathematics, and it has many practical applications in fields such as algebra and geometry.

Frequently Asked Questions

Q: What is the cube root of 128?

A: The cube root of 128 is $\sqrt[3]{128}$ .

Q: How do you rewrite the cube root of 128 in its simplest form?

A: To rewrite the cube root of 128 in its simplest form, you need to break down 128 into its prime factors, apply the power rule of exponents, and simplify the fractional exponent.

Q: What is the final answer to the problem of rewriting $\sqrt[3]{128}$ in its simplest form?

A: The final answer to the problem of rewriting $\sqrt[3]{128}$ in its simplest form is $2^{7/3}$ .

References

[1] "Algebra" by Michael Artin
[2] "Geometry" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman

Q&A: Cube Roots

Q: What is a cube root?

A: A cube root is a type of radical expression that involves the extraction of a cube root of a number. The cube root of a number $x$ is denoted by $\sqrt[3]{x}$ and is defined as the number that, when multiplied by itself twice, gives the original number $x$ .

Q: How do you calculate the cube root of a number?

A: To calculate the cube root of a number, you need to find the number that, when multiplied by itself twice, gives the original number. For example, the cube root of 27 is 3, because $3 \times 3 \times 3 = 27$ .

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

A: A square root is a type of radical expression that involves the extraction of a square root of a number. The square root of a number $x$ is denoted by $\sqrt{x}$ and is defined as the number that, when multiplied by itself, gives the original number $x$ . In contrast, a cube root is a type of radical expression that involves the extraction of a cube root of a number.

Q: How do you simplify a cube root?

A: To simplify a cube root, you need to break down the number inside the radical sign into its prime factors and then apply the power rule of exponents. For example, the cube root of 128 can be simplified as $\sqrt[3]{128} = \sqrt[3]{2^7} = 2^{7/3}$ .

Q: What is the simplest form of a cube root?

A: The simplest form of a cube root is a number that cannot be simplified further. For example, the cube root of 27 is 3, because $3 \times 3 \times 3 = 27$ . This is the simplest form of the cube root of 27.

Q: How do you rewrite a cube root in its simplest form?

A: To rewrite a cube root in its simplest form, you need to break down the number inside the radical sign into its prime factors and then apply the power rule of exponents. For example, the cube root of 128 can be rewritten in its simplest form as $2^{7/3}$ .

Q: What are some common mistakes to avoid when working with cube roots?

A: Some common mistakes to avoid when working with cube roots include:

Not breaking down the number inside the radical sign into its prime factors
Not applying the power rule of exponents
Not simplifying the fractional exponent
Not checking for any common factors that can be canceled out

Q: How do you check if a cube root is in its simplest form?

A: To check if a cube root is in its simplest form, you need to make sure that the number inside the radical sign has been broken down into its prime factors and that the power rule of exponents has been applied. You also need to check for any common factors that can be canceled out.

Q: What are some real-world applications of cube roots?

A: Some real-world applications of cube roots include:

Calculating the volume of a cube
Determining the length of a side of a cube
Finding the height of a cube
Calculating the surface area of a cube

Conclusion

In this article, we have answered some frequently asked questions about cube roots. We have covered topics such as what a cube root is, how to calculate the cube root of a number, and how to simplify a cube root. We have also discussed some common mistakes to avoid when working with cube roots and how to check if a cube root is in its simplest form. Finally, we have explored some real-world applications of cube roots.

References

[1] "Algebra" by Michael Artin
[2] "Geometry" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman