Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. √(x⁵y³)/z²

Verified step by step guidance

1. Simplify the expression by separating the square root of the numerator and the denominator: \( \frac{\sqrt{x^5 y^3}}{z^2} \).
2. Apply the property of square roots: \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). This gives \( \sqrt{x^5} \cdot \sqrt{y^3} \).
3. Simplify \( \sqrt{x^5} \) by expressing it as \( x^{\frac{5}{2}} \). Similarly, simplify \( \sqrt{y^3} \) as \( y^{\frac{3}{2}} \).
4. Combine the simplified expressions: \( x^{\frac{5}{2}} \cdot y^{\frac{3}{2}} \).
5. Write the final expression as \( \frac{x^{\frac{5}{2}} y^{\frac{3}{2}}}{z^2} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. In this context, the expression √(x⁵y³) indicates the square root of the product of x raised to the fifth power and y raised to the third power. Understanding how to manipulate and simplify radical expressions is crucial for performing operations and simplifying the given expression.

Properties of Exponents

The properties of exponents govern how to simplify expressions involving powers. Key rules include the product of powers (a^m * a^n = a^(m+n)) and the power of a power (a^m)^n = a^(m*n). These rules are essential for simplifying the expression √(x⁵y³) by rewriting it in terms of fractional exponents, which can then be simplified further.

Simplifying Fractions

Simplifying fractions involves reducing the numerator and denominator to their simplest form. In the expression √(x⁵y³)/z², understanding how to simplify the radical in the numerator and how it interacts with the denominator is key. This process often includes factoring out common terms and applying the properties of exponents to achieve a more manageable expression.