Simplify (2(VVs-V^2))/(Vs^2-V^2)

The given problem is an algebraic expression that involves simplification. The expression contains radicals and powers of variables, specifically the square root of 's' represented as √s, and the square of 'V' represented as V². The task is to simplify the expression by performing algebraic operations such as multiplication, division, and subtraction on these radicals and powers. The final goal is to reduce the expression to its simplest form, where all like terms have been combined and any common factors in the numerator and the denominator have been canceled out.

$\frac{2 (V \cdot V s - V^{2})}{V s^{2} - V^{2}}$

Answer

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Solution:

Step 1

Multiply the variable $V$ with itself to get $V^{2}$ in the numerator: $\frac{2 (V^{2} s - V^{2})}{V s^{2} - V^{2}}$ .

Step 2

Extract the common factor $V^{2}$ from the terms in the numerator.

Step 2.1

Take $V^{2}$ out of the term $V^{2} s$ : $\frac{2 (V^{2} (s) - V^{2})}{V s^{2} - V^{2}}$ .

Step 2.2

Take $V^{2}$ out of the term $- V^{2}$ : $\frac{2 (V^{2} (s) + V^{2} \cdot (- 1))}{V s^{2} - V^{2}}$ .

Step 2.3

Combine the factored terms in the numerator: $\frac{2 V^{2} (s - 1)}{V s^{2} - V^{2}}$ .

Step 3

Factor out the common variable $V$ from the denominator.

Step 3.1

Extract $V$ from the term $V s^{2}$ : $\frac{2 V^{2} (s - 1)}{V (s^{2}) - V^{2}}$ .

Step 3.2

Extract $V$ from the term $- V^{2}$ : $\frac{2 V^{2} (s - 1)}{V (s^{2}) + V (- V)}$ .

Step 3.3

Combine the factored terms in the denominator: $\frac{2 V^{2} (s - 1)}{V (s^{2} - V)}$ .

Step 4

Cancel out the common factors $V^{2}$ in the numerator and $V$ in the denominator.

Step 4.1

Factor $V$ from the numerator: $\frac{V (2 V (s - 1))}{V (s^{2} - V)}$ .

Step 4.2

Eliminate the common factors.

Step 4.2.1

Remove the common factor $V$ : $\frac{V (2 V (s - 1))}{V (s^{2} - V)}$ .

Step 4.2.2

Simplify the expression: $\frac{2 V (s - 1)}{s^{2} - V}$ .

Knowledge Notes:

The problem involves simplifying a rational expression, which is an algebraic expression that can be represented as the quotient of two polynomials. The steps taken to simplify the expression include:

Multiplication of Variables: Multiplying a variable by itself raises it to a power, which is a basic operation in algebra.
Factoring: This is the process of breaking down an expression into its constituent factors. In this problem, factoring is used to simplify the expression by extracting common factors from the terms.
Cancellation: When the same factor appears in both the numerator and denominator of a fraction, it can be cancelled out, simplifying the expression further. This is based on the property that a fraction remains unchanged if both the numerator and the denominator are multiplied or divided by the same non-zero number.
Combining Like Terms: This involves adding or subtracting terms that have the same variables raised to the same power.
Algebraic Simplification: The final step in simplifying an algebraic expression is to combine all the previous steps and rewrite the expression in its simplest form.

Understanding these concepts is essential for solving algebraic expressions and equations effectively.