Math Problem Statement

\sqrt{x} $$$$ \sqrt{x^2y^2}\sqrt{xy^2}

Solution

Let's break down and simplify each of these expressions:

  1. x\sqrt{x}:

    • This is the square root of xx. It remains as is, unless you specify any further conditions on xx.
  2. x2y2xy2\sqrt{x^2 y^2} \sqrt{xy^2}:

    • First, simplify x2y2\sqrt{x^2 y^2}: x2y2=x2y2=xy\sqrt{x^2 y^2} = \sqrt{x^2} \cdot \sqrt{y^2} = x \cdot y
    • Now simplify xy2\sqrt{xy^2}: xy2=xy2=xy\sqrt{xy^2} = \sqrt{x} \cdot \sqrt{y^2} = \sqrt{x} \cdot y
    • Combining both parts: x2y2xy2=(xy)(xy)=xy2x\sqrt{x^2 y^2} \cdot \sqrt{xy^2} = (x \cdot y) \cdot (\sqrt{x} \cdot y) = x \cdot y^2 \cdot \sqrt{x}
    • So the simplified expression is: xy2xx y^2 \sqrt{x}

Final answer:

xandxy2x\sqrt{x} \quad \text{and} \quad x y^2 \sqrt{x}

Would you like further clarification or details?

Related Questions:

  1. How would this expression change if xx or yy were negative?
  2. What happens when you take the cube root instead of the square root?
  3. Can you simplify expressions involving cube roots in a similar manner?
  4. How does the simplification change when fractional exponents are used instead of radicals?
  5. How would this expression appear if written using exponents rather than square roots?

Tip:

Always break down complex square roots into smaller components to simplify them step by step.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Simplifying Radicals
Square Roots

Formulas

\(\sqrt{x^2} = x\)
\(\sqrt{y^2} = y\)
\(\sqrt{x^2 y^2} = x \cdot y\)
\(\sqrt{xy^2} = \sqrt{x} \cdot y\)

Theorems

Properties of Square Roots

Suitable Grade Level

Grades 9-10