Let's simplify the given expression step by step. The expression is:

$\frac{10 \sqrt{x^7}}{7 \sqrt{x^2}}$

Step 1: Simplify the square roots

• For $\sqrt{x^7}$, recall that $\sqrt{x^7} = x^{7/2} = x^3 \cdot \sqrt{x}$ because $7/2 = 3 + 1/2$.
• For $\sqrt{x^2}$, note that $\sqrt{x^2} = x$.

Thus, the expression becomes:

$\frac{10 \cdot x^3 \cdot \sqrt{x}}{7 \cdot x}$

Step 2: Simplify the fraction

Now divide $x^3 \cdot \sqrt{x}$ by $x$. Recall the property of exponents: $x^a / x^b = x^{a-b}$.

• $x^3 / x = x^{3-1} = x^2$.
• So, the expression becomes:

$\frac{10 \cdot x^2 \cdot \sqrt{x}}{7}$

Step 3: Combine terms

Write the final simplified form:

$\frac{10 x^2 \sqrt{x}}{7}$

Final Answer:

The simplified form of the expression is:

$\frac{10 x^2 \sqrt{x}}{7}$

Do you have any questions about this process, or would you like further clarification?

Related Questions:

1. How do you simplify $\sqrt{x^n}$ for any $n$?
2. What happens when you divide terms with square roots?
3. Can this be rewritten in a form without a square root?
4. How do properties of exponents work with square roots?
5. What is the importance of rationalizing the denominator in expressions like these?

Tip:

Always break down the square root into integer and fractional powers for easier simplification!