Simplify ( square root of 5xy)/(5y)
The problem provided is an algebraic expression simplification exercise. The task is to perform operations to reduce the expression given, which involves a square root of a product of variables ($5xy$) divided by a product of numbers and a variable ($5y$). The goal is to simplify this expression to its most reduced form by applying algebraic rules such as properties of square roots, division, and cancellation of like terms where applicable.
$\frac{\sqrt{5 x y}}{5 y}$
Answer
Solution:
Step 1: Simplify the given expression
Given the expression $\frac{\sqrt{5xy}}{5y}$, we aim to simplify it by reducing it to its simplest form.
Step 2: Factor out common terms
We notice that the term $5y$ in the denominator can be factored out from the square root in the numerator. This is because $\sqrt{5xy} = \sqrt{5y} \cdot \sqrt{x}$.
Step 3: Cancel out common terms
The $\sqrt{5y}$ in the numerator and the $5y$ in the denominator have common terms. We can cancel out the $\sqrt{5y}$ against part of the $5y$ in the denominator.
Step 4: Write down the simplified expression
After canceling out the common terms, we are left with $\frac{\sqrt{x}}{y}$, which is the simplified form of the original expression.
Knowledge Notes:
To simplify the expression $\frac{\sqrt{5xy}}{5y}$, we need to understand several mathematical concepts:
Square Root: The square root of a product $ab$ can be expressed as the product of the square roots of $a$ and $b$, i.e., $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.
Simplification: Simplifying an expression involves reducing it to its simplest form by factoring and canceling out common terms.
Factoring: Factoring is the process of breaking down an expression into its constituent parts that, when multiplied together, give the original expression.
Cancellation: In a fraction, if the same term appears in both the numerator and the denominator, they can be canceled out. This is based on the property that $\frac{a}{a} = 1$ for any non-zero $a$.
Rationalizing the Denominator: While not explicitly used in this problem, it is a common technique in simplifying expressions involving square roots. It involves multiplying the numerator and denominator by a term that will eliminate the square root in the denominator.
By applying these concepts, we can simplify the given expression by factoring out the common terms and canceling them, which leads us to the final simplified result.
