Simplify ( square root of 12x^2)/48
The given problem is an algebraic expression simplification question. It asks to take the square root of the product of the number 12 and the variable x squared (12x^2) and then to divide that result by 48. The goal is to simplify the expression by following the rules of algebraic manipulation, which includes factoring, reducing like terms, and simplifying numerical and variable components.
$\frac{\sqrt{12 x^{2}}}{48}$
Answer
Solution:
Step 1: Simplify the numerator.
Step 1.1: Express $12x^2$ as $(2x)^2 \cdot 3$.
Step 1.1.1: Extract $4$ from $12$. $\frac{\sqrt{4 \cdot 3 x^2}}{48}$
Step 1.1.2: Represent $4$ as $2^2$. $\frac{\sqrt{2^2 \cdot 3 x^2}}{48}$
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:
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.
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 $2^2$.
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).
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.
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.
