Express in simplest radical form.
x, cubed, square root of, x, to the power 7 , end square root

​

The given expression is:

\[
\sqrt[3]{\sqrt{x^7}}
\]

To simplify it, follow these steps:

### Step 1: Rewrite the expression using rational exponents.
We know that:
\[
\sqrt{x^7} = x^{7/2}
\]
Thus, the original expression becomes:
\[
\sqrt[3]{x^{7/2}} = x^{(7/2) \cdot (1/3)}
\]

### Step 2: Simplify the exponent.
\[
\left(\frac{7}{2} \cdot \frac{1}{3}\right) = \frac{7}{6}
\]

### Step 3: Final expression.
The simplified form is:
\[
x^{7/6}
\]

So, the simplest radical form of the given expression is \( x^{7/6} \).

Would you like more details on any part of the process?

Here are five related questions:
1. How do you simplify expressions with fractional exponents?
2. What is the relationship between roots and exponents?
3. How do you simplify higher-order roots of powers?
4. Can you explain the rules of multiplying exponents?
5. What are some real-world applications of radical expressions?

**Tip:** When simplifying radicals, remember that square roots and cube roots can often be expressed as fractional exponents for easier manipulation.

Express in simplest radical form: cube root of the square root of x to the power 7.

Learn how to express the cube root of the square root of x to the power 7 in its simplest radical form. This problem involves key concepts in algebra, including radicals, exponentiation, and the properties of exponents. Suitable for high school students in Grades 9-12.

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