Simplify ( square root of 50x)/(2 square root of 2)
The problem provided is asking for the simplification of a mathematical expression which involves square roots and algebraic terms. To be more specific, the expression takes the square root of the product of 50 and a variable x, then divides this whole square root by another expression consisting of 2 times the square root of 2. The goal is to manipulate and reduce this expression to its simplest form, ideally eliminating the radical from the denominator and simplifying any numerical factors.
$\frac{\sqrt{50 x}}{2 \sqrt{2}}$
Answer
Solution:
Step:1
Merge the square roots in the numerator and denominator into a single square root expression: $\frac{\sqrt{50x/2}}{2}$
Step:2
Simplify the fraction inside the radical.
Step:2.1
Extract the factor of $2$ from $50x$: $\frac{\sqrt{2 \cdot (25x)/2}}{2}$
Step:2.2
Eliminate the common factors inside the radical.
Step:2.2.1
Separate the factor of $2$ in the denominator: $\frac{\sqrt{2 \cdot (25x)/(2 \cdot 1)}}{2}$
Step:2.2.2
Reduce the common factors: $\frac{\sqrt{\cancel{2} \cdot (25x)/\cancel{2} \cdot 1}}{2}$
Step:2.2.3
Reformulate the expression: $\frac{\sqrt{25x/1}}{2}$
Step:2.2.4
Divide $25x$ by $1$: $\frac{\sqrt{25x}}{2}$
Step:3
Refine the square root in the numerator.
Step:3.1
Express $25$ as a square of $5$: $\frac{\sqrt{5^2 \cdot x}}{2}$
Step:3.2
Extract terms from under the square root: $\frac{5\sqrt{x}}{2}$
Knowledge Notes:
To simplify a radical expression, especially when it involves a fraction, you can follow these steps:
Combining Radicals: If you have a fraction under a radical, you can combine them into a single radical expression. This is possible because $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.
Factoring and Reducing: Look for common factors in the numerator and denominator that can be simplified. Factoring helps to identify these common factors.
Simplifying Inside the Radical: Once you've combined the radicals and factored out common terms, you can simplify the expression inside the radical. If there's a perfect square inside the radical, you can take the square root of that term and move it outside the radical.
Simplifying the Numerator: If the numerator is a product of a perfect square and another term, you can simplify it by taking the square root of the perfect square and multiplying it by the square root of the remaining term.
Rationalizing the Denominator: In some cases, you might need to rationalize the denominator (i.e., eliminate the radical from the denominator). However, in this problem, the denominator is already rational.
Final Simplification: The last step is to simplify the expression as much as possible, which might involve dividing terms or further reducing fractions.
In this particular problem, we used the property that the square root of a product is equal to the product of the square roots, which allowed us to simplify the expression significantly. Additionally, we recognized that $25$ is a perfect square, which is why we could take the square root of $25$ and move it outside the radical.
