The problem asks us to simplify the expression $\sqrt{49x^{10}y^8}$.

1. Problem Description

The problem asks us to simplify the expression $\sqrt{49x^{10}y^8}$.

2. Solution Steps

We need to find the square root of $49x^{10}y^8$. We can rewrite the square root as an exponent of $1/2$.

$\sqrt{49x^{10}y^8} = (49x^{10}y^8)^{1/2}$

We can use the power of a product rule: $(ab)^n = a^n b^n$

$(49x^{10}y^8)^{1/2} = 49^{1/2}(x^{10})^{1/2}(y^8)^{1/2}$

Now, we simplify each term.

$49^{1/2} = \sqrt{49} = 7$

For the exponents, we use the power of a power rule: $(a^m)^n = a^{m \cdot n}$

$(x^{10})^{1/2} = x^{10 \cdot (1/2)} = x^5$

$(y^8)^{1/2} = y^{8 \cdot (1/2)} = y^4$

Putting it all together:

$49^{1/2}(x^{10})^{1/2}(y^8)^{1/2} = 7x^5y^4$

3. Final Answer

The simplified expression is $7x^5y^4$.