https://file.aiquickdraw.com/mathbot/file/1725885751589pjhcyeri.jpg

The question in the image asks why \( \sqrt[3]{7} \) is equal to \( 7^{\frac{1}{3}} \). Let's break it down and explain the reasoning behind this.

### Step 1: Definition of \( \sqrt[3]{7} \)
The cube root of 7, \( \sqrt[3]{7} \), is the number that when raised to the power of 3 gives 7. In mathematical terms:
\[
\sqrt[3]{7} = 7^{\frac{1}{3}}
\]
This is the standard way to express a root using exponents. Here, the exponent \( \frac{1}{3} \) represents the cube root.

### Step 2: Understanding the Power of Powers
The problem seems to deal with the equation \( \left( 7^{\frac{1}{3}} \right)^3 \). According to the power of powers rule:
\[
\left( a^m \right)^n = a^{m \cdot n}
\]
In this case:
\[
\left( 7^{\frac{1}{3}} \right)^3 = 7^{\frac{1}{3} \cdot 3} = 7^1 = 7
\]
So, the cube root of \( 7 \) raised to the power of 3 results in \( 7 \), which confirms that \( \sqrt[3]{7} = 7^{\frac{1}{3}} \).

### Step 3: Verifying the Correct Answer
Looking at the options provided:
- The second option is highlighted and shows the correct steps:
  \[
  \left( 7^{\frac{1}{3}} \right)^3 = 7^{1} = 7
  \]
This confirms that the highlighted choice is correct.

### Summary:
The cube root of 7, \( \sqrt[3]{7} \), is equal to \( 7^{\frac{1}{3}} \), and when raised to the power of 3, it gives \( 7 \). This explains why \( \sqrt[3]{7} \) equals \( 7^{\frac{1}{3}} \).

Would you like more details or have any questions about this explanation?

Here are 5 related questions for further practice:
1. What is the general form of expressing any root as a fractional exponent?
2. How would you simplify \( \left( 5^{\frac{1}{4}} \right)^4 \)?
3. What is \( \left( 2^{\frac{1}{2}} \right)^2 \)?
4. How do you express the square root of 16 using exponents?
5. What is the value of \( \sqrt[5]{32} \) expressed as a power of 2?

**Tip:** Remember, fractional exponents follow the same rules as regular exponents, and they are particularly useful for simplifying roots.

Why is the cube root of 7 equal to 7 raised to the power of 1/3?

Understand why the cube root of 7 equals 7 raised to the power of 1/3 using the laws of exponents. Learn about cube roots and exponentiation with a step-by-step breakdown. Suitable for students in Grades 9-11.

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