Comparison of Expressions F and G

Oct 13, 2025 by miodeya.com 57 views

Introduction

In mathematics, we often come across expressions that involve radicals and exponents. These expressions can be simplified using various mathematical operations, and in this article, we will explore two such expressions involving radicals and exponents. We will simplify each expression and determine which one is equal to an integer.

Expression F

The first expression is given by:

F. \frac{\sqrt[3]{32} \sqrt[6]{4}}{\sqrt{2}}

To simplify this expression, we can start by evaluating the cube root of 32 and the sixth root of 4.

\sqrt[3]{32} = \sqrt[3]{2^5} = 2^{\frac{5}{3}}

\sqrt[6]{4} = \sqrt[6]{2^2} = 2^{\frac{1}{3}}

Now, we can substitute these values back into the original expression.

F. \frac{\sqrt[3]{32} \sqrt[6]{4}}{\sqrt{2}} = \frac{2^{\frac{5}{3}} \cdot 2^{\frac{1}{3}}}{\sqrt{2}}

Using the properties of exponents, we can simplify the numerator.

2^{\frac{5}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{5}{3} + \frac{1}{3}} = 2^2

Now, we can substitute this value back into the original expression.

F. \frac{\sqrt[3]{32} \sqrt[6]{4}}{\sqrt{2}} = \frac{2^2}{\sqrt{2}}

To simplify the expression further, we can rationalize the denominator by multiplying both the numerator and the denominator by the square root of 2.

\frac{2^2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{2^2 \sqrt{2}}{2}

Using the properties of exponents, we can simplify the numerator.

2^2 \sqrt{2} = 2^2 \cdot 2^{\frac{1}{2}} = 2^{\frac{5}{2}}

Now, we can substitute this value back into the original expression.

F. \frac{\sqrt[3]{32} \sqrt[6]{4}}{\sqrt{2}} = \frac{2^{\frac{5}{2}}}{2}

Using the properties of exponents, we can simplify the expression further.

\frac{2^{\frac{5}{2}}}{2} = 2^{\frac{5}{2} - 1} = 2^{\frac{3}{2}}

Expression G

The second expression is given by:

G. \frac{\sqrt{8} \sqrt[3]{16}}{\sqrt[6]{2}}

To simplify this expression, we can start by evaluating the square root of 8 and the cube root of 16.

\sqrt{8} = \sqrt{2^3} = 2^{\frac{3}{2}}

\sqrt[3]{16} = \sqrt[3]{2^4} = 2^{\frac{4}{3}}

Now, we can substitute these values back into the original expression.

G. \frac{\sqrt{8} \sqrt[3]{16}}{\sqrt[6]{2}} = \frac{2^{\frac{3}{2}} \cdot 2^{\frac{4}{3}}}{\sqrt[6]{2}}

Using the properties of exponents, we can simplify the numerator.

2^{\frac{3}{2}} \cdot 2^{\frac{4}{3}} = 2^{\frac{3}{2} + \frac{4}{3}} = 2^{\frac{9}{6} + \frac{8}{6}} = 2^{\frac{17}{6}}

Now, we can substitute this value back into the original expression.

G. \frac{\sqrt{8} \sqrt[3]{16}}{\sqrt[6]{2}} = \frac{2^{\frac{17}{6}}}{\sqrt[6]{2}}

To simplify the expression further, we can rationalize the denominator by multiplying both the numerator and the denominator by the sixth root of 2.

\frac{2^{\frac{17}{6}}}{\sqrt[6]{2}} \cdot \frac{\sqrt[6]{2}}{\sqrt[6]{2}} = \frac{2^{\frac{17}{6}} \sqrt[6]{2}}{2^{\frac{1}{6}}}

Using the properties of exponents, we can simplify the numerator.

2^{\frac{17}{6}} \sqrt[6]{2} = 2^{\frac{17}{6}} \cdot 2^{\frac{1}{6}} = 2^{\frac{18}{6}} = 2^3

Now, we can substitute this value back into the original expression.

G. \frac{\sqrt{8} \sqrt[3]{16}}{\sqrt[6]{2}} = \frac{2^3}{2^{\frac{1}{6}}}

Using the properties of exponents, we can simplify the expression further.

\frac{2^3}{2^{\frac{1}{6}}} = 2^{3 - \frac{1}{6}} = 2^{\frac{17}{6}}

Conclusion

In this article, we simplified two expressions involving radicals and exponents. We found that expression F is equal to $2^{\frac{3}{2}}$ , while expression G is equal to $2^{\frac{17}{6}}$ . Since $2^{\frac{3}{2}}$ is an integer, we can conclude that expression F is equal to an integer.

Comparison of Expressions F and G

In this section, we will compare the two expressions and determine which one is equal to an integer.

As we can see from the previous section, expression F is equal to $2^{\frac{3}{2}}$ , while expression G is equal to $2^{\frac{17}{6}}$ . Since $2^{\frac{3}{2}}$ is an integer, we can conclude that expression F is equal to an integer.

On the other hand, expression G is equal to $2^{\frac{17}{6}}$ , which is not an integer. Therefore, we can conclude that expression G is not equal to an integer.

Final Answer

Based on our analysis, we can conclude that expression F is equal to an integer, while expression G is not equal to an integer.

Key Takeaways

Expression F is equal to $2^{\frac{3}{2}}$ , which is an integer.
Expression G is equal to $2^{\frac{17}{6}}$ , which is not an integer.
To determine which expression is equal to an integer, we need to simplify the expressions and evaluate their values.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by Michael Spivak

Additional Resources

[1] Khan Academy: Algebra and Trigonometry
[2] MIT OpenCourseWare: Calculus

Introduction

In our previous article, we simplified two expressions involving radicals and exponents. We found that expression F is equal to $2^{\frac{3}{2}}$ , while expression G is equal to $2^{\frac{17}{6}}$ . Since $2^{\frac{3}{2}}$ is an integer, we can conclude that expression F is equal to an integer. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the difference between expression F and expression G?

A: The main difference between expression F and expression G is the value of the exponents. Expression F has an exponent of $\frac{3}{2}$ , while expression G has an exponent of $\frac{17}{6}$ . This difference in exponents results in different values for the two expressions.

Q: Why is expression F equal to an integer, while expression G is not?

A: Expression F is equal to an integer because its exponent is a rational number that can be simplified to an integer. On the other hand, expression G has an exponent that cannot be simplified to an integer, resulting in a non-integer value.

Q: Can you provide more examples of expressions that are equal to integers?

A: Yes, here are a few examples of expressions that are equal to integers:

$\sqrt{16} = 4$
$\sqrt[3]{27} = 3$
$\sqrt[4]{16} = 2$

Q: Can you provide more examples of expressions that are not equal to integers?

A: Yes, here are a few examples of expressions that are not equal to integers:

$\sqrt{2}$
$\sqrt[3]{2}$
$\sqrt[4]{2}$

Q: How can I determine if an expression is equal to an integer?

A: To determine if an expression is equal to an integer, you can simplify the expression and evaluate its value. If the value is a rational number that can be simplified to an integer, then the expression is equal to an integer.

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

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

Not following the order of operations
Not simplifying the expression correctly
Not evaluating the value of the expression

Q: Can you provide some tips for simplifying expressions?

A: Yes, here are some tips for simplifying expressions:

Follow the order of operations
Simplify the expression step by step
Evaluate the value of the expression

Conclusion

In this article, we answered some frequently asked questions related to the topic of expressions that are equal to integers. We provided examples of expressions that are equal to integers and expressions that are not equal to integers. We also provided tips for simplifying expressions and common mistakes to avoid.

Key Takeaways

Expression F is equal to $2^{\frac{3}{2}}$ , which is an integer.
Expression G is equal to $2^{\frac{17}{6}}$ , which is not an integer.
To determine if an expression is equal to an integer, you can simplify the expression and evaluate its value.
Some common mistakes to avoid when simplifying expressions include not following the order of operations, not simplifying the expression correctly, and not evaluating the value of the expression.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by Michael Spivak

Additional Resources

[1] Khan Academy: Algebra and Trigonometry
[2] MIT OpenCourseWare: Calculus