Simplify ( square root of 12x^2)/48

Solution:

Step 1: Simplify the numerator.

• Step 1.1: Express $12x^2$ as $(2x{)}^2\cdot 3$.

• Step 1.1.1: Extract $4$ from $12$. $\frac{\sqrt{4\cdot 3x^2}}{48}$

• Step 1.1.2: Represent $4$ as $2^2$. $\frac{\sqrt{2^2\cdot 3x^2}}{48}$

• Step 1.1.3: Rearrange $3$. $\frac{\sqrt{2^2{x}^2\cdot 3}}{48}$

• Step 1.1.4: Rewrite $2^2{x}^2$ as $(2x{)}^2$. $\frac{\sqrt{(2x{)}^2\cdot 3}}{48}$

• Step 1.2: Extract terms from under the square root. $\frac{2x\sqrt{3}}{48}$

Step 2: Reduce the fraction by eliminating common factors.

• Step 2.1: Factor out $2$ from $2x\sqrt{3}$. $\frac{2(x\sqrt{3})}{48}$

• Step 2.2: Eliminate common factors.

  • Step 2.2.1: Factor out $2$ from $48$. $\frac{2(x\sqrt{3})}{2\cdot 24}$
  • Step 2.2.2: Cancel out the common factor of $2$. $\frac{\cancel{2}(x\sqrt{3})}{\cancel{2}\cdot 24}$
  • Step 2.2.3: Simplify the expression. $\frac{x\sqrt{3}}{24}$

Knowledge Notes:

To simplify the expression $\frac{\sqrt{12x^2}}{48}$, we apply several mathematical principles and properties:

1. Radical Simplification: The square root of a product is equal to the product of the square roots of each factor, provided that all quantities under the square root are non-negative.

2. Factoring: We can factor numbers and expressions to reveal common factors that can be simplified. For example, $12x^2$ can be factored into $4\cdot 3x^2$, and $4$ can be further expressed as $2^2$.

3. Square Root of a Square: The square root of a squared number or variable, such as $\sqrt{(2x{)}^2}$, simplifies to the absolute value of the original number or variable, which in this case is $2x$ (assuming $x$ is non-negative).

4. Fraction Reduction: To simplify fractions, we look for common factors in the numerator and the denominator that can be divided out. In this case, we divide both the numerator and the denominator by $2$ to simplify the fraction.

5. Algebraic Manipulation: Throughout the process, we use algebraic manipulation to rearrange and combine terms in a way that makes simplification possible.

By applying these principles step by step, we can simplify the original expression to its most reduced form.