Simplify square root of (72x^3)/(2x)

Solution:

Simplification Process:

Step 1: Simplify the fraction $\frac{72x^3}{2x}$ by removing common factors.

Step 1.1: Extract the factor of $2$ from the numerator: $\sqrt{\frac{2(36x^3)}{2x}}$.

Step 1.2: Extract the factor of $2$ from the denominator: $\sqrt{\frac{2(36x^3)}{2(x)}}$.

Step 1.3: Eliminate the common factor of $2$: $\sqrt{\frac{\cancel{2}(36x^3)}{\cancel{2}x}}$.

Step 1.4: Rewrite the simplified expression: $\sqrt{\frac{36x^3}{x}}$.

Step 2: Further simplify by canceling out common $x$ terms.

Step 2.1: Factor out $x$ from the numerator: $\sqrt{\frac{x(36x^2)}{x}}$.

Step 2.2: Proceed to cancel common factors.

Step 2.2.1: Express the denominator as $x^1$: $\sqrt{\frac{x(36x^2)}{x^1}}$.

Step 2.2.2: Factor out $x$ from the denominator: $\sqrt{\frac{x(36x^2)}{x\cdot 1}}$.

Step 2.2.3: Cancel the common $x$ factor: $\sqrt{\frac{\cancel{x}(36x^2)}{\cancel{x}\cdot 1}}$.

Step 2.2.4: Rewrite the expression: $\sqrt{\frac{36x^2}{1}}$.

Step 2.2.5: Simplify the division by $1$: $\sqrt{36x^2}$.

Step 3: Express $36x^2$ as a perfect square: $\sqrt{(6x{)}^2}$.

Step 4: Extract the square root of the perfect square, assuming $x$ is a positive real number: $6x$.

Knowledge Notes:

To simplify the square root of a fraction, you can follow these steps:

1. Factorization: Break down both the numerator and the denominator into their prime factors or common factors to simplify the fraction.

2. Simplification: Cancel out any common factors in the numerator and the denominator.

3. Square Roots: For terms under a square root, rewrite them as perfect squares if possible. This allows you to take the square root more easily.

4. Radical Rules: Remember that the square root of a fraction is the square root of the numerator divided by the square root of the denominator. Also, the square root of a product of terms is the product of the square roots of the individual terms.

5. Assumptions: When pulling terms out from under the radical, it is typically assumed that the variables represent positive real numbers to avoid dealing with complex numbers or negative roots.

6. Algebraic Manipulation: Use algebraic rules to combine like terms and simplify expressions. This includes understanding how to handle exponents and roots.

7. Perfect Squares: Recognize perfect squares such as $1,4,9,16,25,36,...$ which can be square rooted easily.

By applying these principles, you can simplify expressions involving square roots and fractions effectively.