Simplify square root of 5/(10x)
Explanation: You are being asked to simplify the mathematical expression that involves taking the square root of a fraction. The fraction in question is 5 divided by the product of 10 and a variable x. The simplification process would likely involve applying rules of simplifying square roots and fractions, as well as algebraic manipulation to express the result in the simplest form possible.
$\sqrt{\frac{5}{10 x}}$
Answer
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:
Simplification of Fractions: We reduced the fraction by canceling out common factors in the numerator and denominator.
Extraction of Factors: We factored out common terms from both the numerator and the denominator to simplify the expression.
Square Roots: We separated the square root of a fraction into the square root of the numerator over the square root of the denominator.
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.
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.
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.
