Let's simplify the given expression step-by-step:

The expression given is:

$5x^3y \sqrt{4x^5y^7}$

Step 1: Simplify the expression inside the square root

Inside the square root, we have:

$4x^5y^7$

The square root of each term can be simplified individually:

$\sqrt{4} = 2, \quad \sqrt{x^5} = x^{\frac{5}{2}}, \quad \sqrt{y^7} = y^{\frac{7}{2}}$

Putting it all together, we get:

$\sqrt{4x^5y^7} = 2x^{\frac{5}{2}}y^{\frac{7}{2}}$

Step 2: Multiply the simplified square root by the remaining terms

Now, multiply the simplified square root by the remaining part of the expression:

$5x^3y \times 2x^{\frac{5}{2}}y^{\frac{7}{2}}$

This can be simplified further by multiplying the coefficients and adding the exponents of like bases:

• Coefficient: $5 \times 2 = 10$
• Exponent of $x$: $3 + \frac{5}{2} = \frac{6}{2} + \frac{5}{2} = \frac{11}{2}$
• Exponent of $y$: $1 + \frac{7}{2} = \frac{2}{2} + \frac{7}{2} = \frac{9}{2}$

So, the simplified expression is:

$10x^{\frac{11}{2}}y^{\frac{9}{2}}$

Final Simplified Form

$\boxed{10x^{\frac{11}{2}}y^{\frac{9}{2}}}$

Would you like more details or have any questions?

Related Questions:

1. How do you simplify expressions with fractional exponents?
2. What are the rules for multiplying exponents with the same base?
3. How do you simplify square roots of variables with exponents?
4. Can fractional exponents be converted back to radical form?
5. What is the general process for simplifying radical expressions with coefficients?

Tip: When multiplying terms with exponents, remember to add the exponents if the bases are the same!