Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. $\sqrt{\frac{x^5{y}^3}{z^2}}$

Verified step by step guidance

1. Rewrite the expression using exponents instead of radicals. Recall that the square root of a variable is the same as raising it to the power of 1/2. So, rewrite \(\sqrt{x^{5} y^{3}}\) as \((x^{5} y^{3})^{\frac{1}{2}}\).
2. Apply the power of a product rule: \((ab)^m = a^m b^m\). This means \((x^{5} y^{3})^{\frac{1}{2}} = x^{5 \cdot \frac{1}{2}} y^{3 \cdot \frac{1}{2}}\).
3. Simplify the exponents by multiplying: \(x^{\frac{5}{2}} y^{\frac{3}{2}}\).
4. Rewrite the entire expression as \(\frac{x^{\frac{5}{2}} y^{\frac{3}{2}}}{z^{2}}\) since the denominator \(z^{2}\) remains unchanged.
5. Check if any further simplification is possible by expressing fractional exponents as radicals or by factoring exponents, depending on the context or instructions.

Key Concepts

Properties of Exponents

Understanding how to manipulate exponents is essential for simplifying expressions. This includes rules like multiplying powers with the same base, dividing powers, and raising powers to powers. For example, x^a / x^b = x^(a-b) and (x^a)^b = x^(ab).

Simplifying Radicals

Simplifying radicals involves expressing the radicand as a product of perfect squares (or higher powers) and other factors to simplify the root. For example, √(x⁵) can be rewritten as x²√x by separating powers into even and odd exponents.

Division of Expressions with Variables

When dividing expressions with variables, apply exponent rules to subtract exponents of like bases in numerator and denominator. Also, keep track of variables under radicals and outside to simplify the entire expression correctly.