Simplify square root of (6m^4n^5)/(12m^5n^4)

Solution:

Simplification Process:

Step 1: Simplify the fraction $\frac{6m^4{n}^5}{12m^5{n}^4}$ by reducing common terms.

Step 1.1: Extract the factor of 6 from the numerator $\sqrt{\frac{6(m^4{n}^5)}{12m^5{n}^4}}$.

Step 1.2: Extract the factor of 6 from the denominator $\sqrt{\frac{6(m^4{n}^5)}{6(2m^5{n}^4)}}$.

Step 1.3: Eliminate the common factor of 6 $\sqrt{\frac{\cancel{6}(m^4{n}^5)}{\cancel{6}(2m^5{n}^4)}}$.

Step 1.4: Present the simplified fraction $\sqrt{\frac{m^4{n}^5}{2m^5{n}^4}}$.

Step 2: Remove common powers of $m$.

Step 2.1: Factor out $m^4$ from the numerator $\sqrt{\frac{m^4(n^5)}{2m^5{n}^4}}$.

Step 2.2: Cancel out common powers of $m$.

Step 2.2.1: Factor out $m^4$ from the denominator $\sqrt{\frac{m^4(n^5)}{m^4(2m{n}^4)}}$.

Step 2.2.2: Cancel the common power of $m^4$ $\sqrt{\frac{\cancel{m^4}{n}^5}{\cancel{m^4}(2m{n}^4)}}$.

Step 2.2.3: Represent the new expression $\sqrt{\frac{n^5}{2m{n}^4}}$.

Step 3: Remove common powers of $n$.

Step 3.1: Factor out $n^4$ from the numerator $\sqrt{\frac{n^{4}n}{2m{n}^4}}$.

Step 3.2: Cancel out common powers of $n$.

Step 3.2.1: Factor out $n^4$ from the denominator $\sqrt{\frac{n^{4}n}{n^4(2m)}}$.

Step 3.2.2: Cancel the common power of $n^4$ $\sqrt{\frac{\cancel{n^4}n}{\cancel{n^4}(2m)}}$.

Step 3.2.3: Present the simplified expression $\sqrt{\frac{n}{2m}}$.

Step 4: Rewrite the square root of a fraction as a fraction of square roots $\frac{\sqrt{n}}{\sqrt{2m}}$.

Step 5: Rationalize the denominator by multiplying by $\frac{\sqrt{2m}}{\sqrt{2m}}$.

Step 6: Simplify the denominator.

Step 6.1: Multiply the numerator and denominator by $\sqrt{2m}$ $\frac{\sqrt{n}\sqrt{2m}}{\sqrt{2m}\sqrt{2m}}$.

Step 6.2: Apply the power of a power rule $\frac{\sqrt{n}\sqrt{2m}}{(\sqrt{2m}{)}^1\sqrt{2m}}$.

Step 6.3: Combine the powers in the denominator $\frac{\sqrt{n}\sqrt{2m}}{(\sqrt{2m}{)}^{1+1}}$.

Step 6.4: Simplify the exponent $\frac{\sqrt{n}\sqrt{2m}}{(\sqrt{2m}{)}^2}$.

Step 6.5: Rewrite the square of a square root as the original number $\frac{\sqrt{n}\sqrt{2m}}{2m}$.

Step 7: Combine the radicals in the numerator using the product rule $\frac{\sqrt{2nm}}{2m}$.

Step 8: Reorder the factors inside the radical for clarity $\frac{\sqrt{2mn}}{2m}$.

Knowledge Notes:

The problem involves simplifying a radical expression with a fraction under the square root. The process requires knowledge of several algebraic rules and properties:

1. Common Factor Reduction: When the same factor appears in both the numerator and the denominator, it can be cancelled out.

2. Exponent Rules: When the same base is raised to different exponents, the powers can be combined by adding or subtracting the exponents depending on the operation (multiplication or division).

3. Square Root of a Fraction: The square root of a fraction can be expressed as the fraction of the square roots of the numerator and the denominator.

4. Rationalizing the Denominator: This involves removing the square root from the denominator of a fraction by multiplying the numerator and denominator by an appropriate form of 1 (such as $\frac{\sqrt{2m}}{\sqrt{2m}}$).

5. Product Rule for Radicals: $\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}$.

6. Power of a Power Rule: $(a^m{)}^n=a^{mn}$.

7. Square of a Square Root: $(\sqrt{a}{)}^2=a$.

The solution applies these rules step by step to simplify the given radical expression.