Simplify square root of (72x^3)/(2x)
The given problem requires the simplification of a radical expression. Specifically, you are asked to simplify the square root of a fraction. The numerator of this fraction is 72 times x cubed (72x^3), and the denominator is 2 times x (2x). The task is to find a simplified form of this expression by performing valid algebraic operations within the square root, which could include simplifying the fraction by dividing the terms and/or factoring common factors from the radicand (the expression inside the square root), and then applying properties of square roots to further simplify the expression.
$\sqrt{\frac{72 x^{3}}{2 x}}$
Answer
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:
Factorization: Break down both the numerator and the denominator into their prime factors or common factors to simplify the fraction.
Simplification: Cancel out any common factors in the numerator and the denominator.
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.
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.
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.
Algebraic Manipulation: Use algebraic rules to combine like terms and simplify expressions. This includes understanding how to handle exponents and roots.
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.
