Simplify square root of ((0.487x-35.999)^2)/(6-2)

Solution:

Step 1: Extract the common factor

Extract the factor $0.001$ from $0.487x - 35.999$.

Step 1.1: Factor out from the variable term

Extract $0.001$ from $0.487x$: $\sqrt{\frac{(\left(0.001 \cdot 487x\right) - 35.999)^2}{6 - 2}}$

Step 1.2: Factor out from the constant term

Extract $0.001$ from $-35.999$: $\sqrt{\frac{(\left(0.001 \cdot 487x\right) + \left(0.001 \cdot -35999\right))^2}{6 - 2}}$

Step 1.3: Combine the factored terms

Combine the factored expressions: $\sqrt{\frac{(\left(0.001 \cdot (487x - 35999)\right))^2}{6 - 2}}$

Step 2: Simplify the squared expression

Step 2.1: Apply the square of a product rule

Square the factor $0.001$: $\sqrt{\frac{(0.001^2 \cdot (487x - 35999)^2)}{6 - 2}}$

Step 2.2: Simplify the denominator

Subtract $2$ from $6$: $\sqrt{\frac{(0.001^2 \cdot (487x - 35999)^2)}{4}}$

Step 2.3: Square the factor

Square $0.001$: $\sqrt{\frac{0.000001 \cdot (487x - 35999)^2}{4}}$

Step 3: Extract the squared factor

Extract $0.000001$ from the squared term: $\sqrt{\frac{0.000001 \cdot ((487x - 35999)^2)}{4}}$

Step 4: Factor out the denominator

Factor $4$ from the denominator: $\sqrt{\frac{0.000001 \cdot ((487x - 35999)^2)}{4 \cdot 1}}$

Step 5: Separate the fractions

Separate the fraction into two parts: $\sqrt{\frac{0.000001}{4} \cdot \frac{(487x - 35999)^2}{1}}$

Step 6: Divide the factors

Divide $0.000001$ by $4$: $\sqrt{0.00000025 \cdot \frac{(487x - 35999)^2}{1}}$

Step 7: Simplify by canceling common factors

Step 7.1: Factor out from the numerator

Factor $0.00000025$ from the numerator: $\sqrt{0.00000025 \cdot \frac{(487x - 35999)^2}{0.00000025 \cdot 4000000}}$

Step 7.2: Cancel the common factors

Cancel out $0.00000025$: $\sqrt{\frac{(487x - 35999)^2}{4000000}}$

Step 7.3: Rewrite the simplified expression

Rewrite the expression without the canceled factor: $\sqrt{\frac{(487x - 35999)^2}{4000000}}$

Step 8: Express the denominator as a square

Rewrite $4000000$ as $(2000)^2$: $\sqrt{\frac{(487x - 35999)^2}{(2000)^2}}$

Step 9: Rewrite as a single fraction under the radical

Combine the terms under the radical: $\sqrt{\left(\frac{487x - 35999}{2000}\right)^2}$

Step 10: Simplify the square root

Simplify the radical assuming positive real numbers: $\frac{487x - 35999}{2000}$

Knowledge Notes:

The problem involves simplifying a square root of a fraction where the numerator is a squared term and the denominator is a simple subtraction. The steps taken to simplify the expression involve factoring out common terms, applying the square of a product rule, simplifying the denominator, separating fractions, and canceling common factors.

Relevant knowledge points include:

• Factoring: The process of breaking down an expression into its constituent factors.

• Square of a product rule: $(ab)^2 = a^2b^2$, which is used to simplify the square of a product of two terms.

• Simplifying fractions: The process of reducing the numerator and denominator to their simplest form by canceling common factors.

• Square roots: The operation of finding a number which, when multiplied by itself, gives the original number. When a square root is applied to a squared expression, it simplifies to the base of the square, provided that the base represents a positive real number.

In this problem, the square root and the squared term cancel each other out, leaving a simplified fraction. The final result is obtained by recognizing that the square root of a square cancels out, assuming we are dealing with positive real numbers.