Simplify square root of 5/(10x)

Solution:

Step 1: Simplify the Fraction

Reduce the fraction $\frac{5}{10x}$ by eliminating common factors.

Step 1.1: Extract the Factor from the Numerator

Extract the factor $5$ from the numerator: $\sqrt{\frac{5\cdot 1}{10x}}$.

Step 1.2: Extract the Factor from the Denominator

Extract the factor $5$ from the denominator: $\sqrt{\frac{5\cdot 1}{5\cdot 2x}}$.

Step 1.3: Eliminate the Common Factor

Eliminate the common factor of $5$: $\sqrt{\frac{\cancel{5}\cdot 1}{\cancel{5}\cdot 2x}}$.

Step 1.4: Finalize the Simplified Expression

The simplified expression is: $\sqrt{\frac{1}{2x}}$.

Step 2: Separate the Square Root

Rewrite the square root as a fraction of square roots: $\frac{\sqrt{1}}{\sqrt{2x}}$.

Step 3: Simplify the Numerator

Since the square root of $1$ is $1$, the expression becomes: $\frac{1}{\sqrt{2x}}$.

Step 4: Rationalize the Denominator

Multiply the expression by $\frac{\sqrt{2x}}{\sqrt{2x}}$ to rationalize the denominator.

Step 5: Simplify the Denominator

Step 5.1: Multiply the Numerator and Denominator

Multiply both the numerator and denominator by $\sqrt{2x}$: $\frac{\sqrt{2x}}{\sqrt{2x}\cdot \sqrt{2x}}$.

Step 5.2: Apply the Power of One

Raise $\sqrt{2x}$ to the power of one: $\frac{\sqrt{2x}}{(\sqrt{2x}{)}^1\cdot \sqrt{2x}}$.

Step 5.3: Repeat the Power of One

Repeat the power of one for the second $\sqrt{2x}$: $\frac{\sqrt{2x}}{(\sqrt{2x}{)}^1\cdot (\sqrt{2x}{)}^1}$.

Step 5.4: Combine the Exponents

Use the power rule to combine the exponents: $\frac{\sqrt{2x}}{(\sqrt{2x}{)}^{1+1}}$.

Step 5.5: Sum the Exponents

Add the exponents together: $\frac{\sqrt{2x}}{(\sqrt{2x}{)}^2}$.

Step 5.6: Simplify the Square Root

Step 5.6.1: Rewrite the Square Root

Rewrite $\sqrt{2x}$ as $(2x{)}^{\frac{1}{2}}$: $\frac{\sqrt{2x}}{((2x{)}^{\frac{1}{2}}{)}^2}$.

Step 5.6.2: Apply the Power Rule

Apply the power rule to multiply the exponents: $\frac{\sqrt{2x}}{(2x{)}^{\frac{1}{2}\cdot 2}}$.

Step 5.6.3: Combine the Fractional Exponents

Combine the fractional exponents: $\frac{\sqrt{2x}}{(2x{)}^{\frac{2}{2}}}$.

Step 5.6.4: Cancel the Common Factor

Cancel out the common factor of $2$: $\frac{\sqrt{2x}}{(2x{)}^{\frac{\cancel{2}}{\cancel{2}}}}$.

Step 5.6.5: Final Simplification

The final simplified expression is: $\frac{\sqrt{2x}}{2x}$.

Knowledge Notes:

To solve the given problem, we applied several mathematical concepts:

1. Simplification of Fractions: We reduced the fraction by canceling out common factors in the numerator and denominator.

2. Extraction of Factors: We factored out common terms from both the numerator and the denominator to simplify the expression.

3. Square Roots: We separated the square root of a fraction into the square root of the numerator over the square root of the denominator.

4. Rationalizing the Denominator: To eliminate the square root from the denominator, we multiplied the fraction by a form of one that contains the square root, thus rationalizing the denominator.

5. Properties of Exponents: We used the power rule for exponents, which states that $a^m\cdot a^n=a^{m+n}$, to combine and simplify the expression.

6. Simplifying Square Roots: We rewrote the square root as a power, applied the power rule, and then simplified the resulting expression.

These steps are essential for manipulating algebraic expressions, especially when dealing with square roots and rational expressions.