
$\frac{\sqrt[5]{x^6.\sqrt{x^7}}}{\sqrt[4]{x^2.\sqrt[5]{x}}}$
5√x6.√x7
4√x2.5√x

We are tasked with simplifying the expression:

\[
\frac{\sqrt[5]{x^6 \cdot \sqrt{x^7}}}{\sqrt[4]{x^2 \cdot \sqrt[5]{x}}}
\]

### Step-by-step simplification:

#### Step 1: Simplify the radicands inside the radicals.

- **Numerator:** $\sqrt[5]{x^6 \cdot \sqrt{x^7}}$

The term $\sqrt{x^7}$ can be rewritten as $x^{7/2}$, since $\sqrt{x^7} = x^{7/2}$. Therefore, we have:

\[
\sqrt[5]{x^6 \cdot x^{7/2}} = \sqrt[5]{x^{6 + 7/2}} = \sqrt[5]{x^{19/2}}
\]

- **Denominator:** $\sqrt[4]{x^2 \cdot \sqrt[5]{x}}$

Similarly, $\sqrt[5]{x}$ is equivalent to $x^{1/5}$, so:

\[
\sqrt[4]{x^2 \cdot x^{1/5}} = \sqrt[4]{x^{2 + 1/5}} = \sqrt[4]{x^{11/5}}
\]

#### Step 2: Apply the fractional exponents for the radicals.

- The fifth root of $x^{19/2}$ is $x^{(19/2) \cdot (1/5)} = x^{19/10}$.
- The fourth root of $x^{11/5}$ is $x^{(11/5) \cdot (1/4)} = x^{11/20}$.

#### Step 3: Simplify the quotient of exponents.

Now, the expression becomes:

\[
\frac{x^{19/10}}{x^{11/20}}
\]

When dividing powers with the same base, subtract the exponents:

\[
x^{\frac{19}{10} - \frac{11}{20}} = x^{\frac{38}{20} - \frac{11}{20}} = x^{\frac{27}{20}}
\]

### Final Answer:
The simplified form of the given expression is:

\[
x^{\frac{27}{20}}
\]

This is the final result.

---

Would you like a detailed explanation of any specific step?

Here are five related questions for practice:
1. How do you simplify expressions involving nested radicals?
2. What are the rules for simplifying fractional exponents?
3. How do you convert roots into fractional exponents?
4. How do you handle the multiplication of terms with exponents?
5. How do you simplify an expression involving the division of exponents with the same base?

**Tip:** When simplifying radicals with variables, always convert the roots into fractional exponents first for easier manipulation.

$\frac{\sqrt[5]{x^6 \cdot \sqrt{x^7}}}{\sqrt[4]{x^2 \cdot \sqrt[5]{x}}}$

Learn how to simplify the expression $\frac{\sqrt[5]{x^6 \cdot \sqrt{x^7}}}{\sqrt[4]{x^2 \cdot \sqrt[5]{x}}}$ with a step-by-step breakdown. This problem involves simplifying nested radicals and converting them to fractional exponents, applying the laws of exponents. Suitable for students in Grades 11-12, exploring advanced algebraic concepts.

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