How do I solve \sqrt{\frac{33xy^{3}}{\sqrt{3x}}}? I am getting so confused and I need help please!
To solve the expression \sqrt{\frac{33xy^{3}}{\sqrt{3x}}}, we can follow these steps:
Step 1: Simplify within the fraction
- Simplify the square root within the fraction by using the property \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}.
- The expression becomes \sqrt{\frac{33xy^{3}}{\sqrt{3}\sqrt{x}}}.
Step 2: Simplify the denominator
- Since \sqrt{3}\sqrt{x} = \sqrt{3x}, the expression becomes \sqrt{\frac{33xy^{3}}{\sqrt{3x}}}.
Step 3: Simplify the numerator
- The expression can be simplified by applying the property \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}.
- Thus, we have \frac{\sqrt{33xy^{3}}}{\sqrt{\sqrt{3x}}}.
Step 4: Simplify the square roots
- Simplify the square root of each term separately:
- The square root of 33 can be written as \sqrt{33}.
- The square root of x can be written as \sqrt{x}.
- The square root of y^3 can be written as \sqrt{y}\sqrt{y^2} = \sqrt{y}\cdot y.
- The square root of \sqrt{3x} can be written as \sqrt[4]{3x}.
- Combining these square roots, we get \frac{\sqrt{33xy}}{\sqrt[4]{3x}}.
Step 5: Simplify the expression (if needed)
- If there are any further possibilities for simplification or if any restrictions are given, apply them to simplify the expression further.
Keep in mind that the simplified expression may depend on any given restrictions on the variables x and y.