https://file.aiquickdraw.com/mathbot/file/17266305141494awty5vk.jpg

The problem in the image asks which of the following shows that \( \sqrt[4]{5} = 5^{\frac{1}{4}} \).

Let's break it down:

We are essentially verifying the equivalence:

\[
\sqrt[4]{5} = 5^{\frac{1}{4}}
\]

This equation means that raising \( 5^{\frac{1}{4}} \) to the power of 4 should give us 5.

Now let's check each option:

### Option A:
\[
\left(5^{\frac{1}{4}}\right)^4 = 5^{\frac{1}{4}} \cdot 5^{\frac{1}{4}} \cdot 5^{\frac{1}{4}} \cdot 5^{\frac{1}{4}} = 5^{\frac{4}{4}} = 5^1 = 5
\]
This shows the correct simplification. \( 5^1 = 5 \).

### Option B:
This also simplifies as follows:
\[
\left(5^{\frac{1}{4}}\right)^4 = 5 \cdot \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right) = 5 \cdot \frac{4}{4} = 5
\]
Correct.

### Option C:
\[
\left(5^{\frac{1}{4}}\right)^4 = 4 \cdot 5^{\frac{1}{4}} = 4 \cdot 5
\]
This is incorrect, because \( 4 \cdot 5 = 20 \), not 5.

### Option D:
\[
\left(5^{\frac{1}{4}}\right)^4 = 5^{\frac{1}{4}} \cdot 5^{\frac{1}{4}} \cdot 5^{\frac{1}{4}} \cdot 5^{\frac{1}{4}} = 5^{\frac{4}{4}} = 5^1 = 5
\]
This is also correct, just like option A.

### Conclusion:
Options **A**, **B**, and **D** correctly show that \( \sqrt[4]{5} = 5^{\frac{1}{4}} \). However, **C** is incorrect.

Would you like more details, or do you have any questions?

Here are 5 related questions for practice:
1. How does exponentiation distribute over multiplication?
2. What is the general rule for simplifying \( a^{\frac{m}{n}} \)?
3. How can we express roots using fractional exponents?
4. Why does multiplying powers with the same base add their exponents?
5. What is the value of \( \sqrt[3]{8} \) expressed as a fractional exponent?

**Tip**: When simplifying expressions involving exponents, always remember that \( a^m \cdot a^n = a^{m+n} \).

Which of the following shows that the fourth root of 5 equals 5 raised to the power of one-fourth?

Explore how the fourth root of 5 is equivalent to 5 raised to the power of one-fourth. This question demonstrates key concepts in exponents and roots, breaking down multiple-choice options for better understanding. Suitable for high school students in Grades 9-12.

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