Simplify square root of (6m^4n^5)/(12m^5n^4)
The question asks for the simplification of a mathematical expression involving square roots and variables with exponents. Specifically, you are required to simplify the square root of a fraction where the numerator is 6 times m raised to the power of 4 times n raised to the power of 5, and the denominator is 12 times m raised to the power of 5 times n raised to the power of 4. The simplification should follow the rules of exponents and radical expressions to reduce the fraction to its simplest form.
$\sqrt{\frac{6 m^{4} n^{5}}{12 m^{5} n^{4}}}$
Answer
Solution:
Simplification Process:
Step 1: Simplify the fraction $\frac{6m^4n^5}{12m^5n^4}$ by reducing common terms.
Step 1.1: Extract the factor of 6 from the numerator $\sqrt{\frac{6(m^4n^5)}{12m^5n^4}}$.
Step 1.2: Extract the factor of 6 from the denominator $\sqrt{\frac{6(m^4n^5)}{6(2m^5n^4)}}$.
Step 1.3: Eliminate the common factor of 6 $\sqrt{\frac{\cancel{6}(m^4n^5)}{\cancel{6}(2m^5n^4)}}$.
Step 1.4: Present the simplified fraction $\sqrt{\frac{m^4n^5}{2m^5n^4}}$.
Step 2: Remove common powers of $m$.
Step 2.1: Factor out $m^4$ from the numerator $\sqrt{\frac{m^4(n^5)}{2m^5n^4}}$.
Step 2.2: Cancel out common powers of $m$.
Step 2.2.1: Factor out $m^4$ from the denominator $\sqrt{\frac{m^4(n^5)}{m^4(2mn^4)}}$.
Step 2.2.2: Cancel the common power of $m^4$ $\sqrt{\frac{\cancel{m^4}n^5}{\cancel{m^4}(2mn^4)}}$.
Step 2.2.3: Represent the new expression $\sqrt{\frac{n^5}{2mn^4}}$.
Step 3: Remove common powers of $n$.
Step 3.1: Factor out $n^4$ from the numerator $\sqrt{\frac{n^4n}{2mn^4}}$.
Step 3.2: Cancel out common powers of $n$.
Step 3.2.1: Factor out $n^4$ from the denominator $\sqrt{\frac{n^4n}{n^4(2m)}}$.
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:
Common Factor Reduction: When the same factor appears in both the numerator and the denominator, it can be cancelled out.
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).
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.
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 $\frac{\sqrt{2m}}{\sqrt{2m}}$).
Product Rule for Radicals: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.
Power of a Power Rule: $(a^m)^n = a^{mn}$.
Square of a Square Root: $(\sqrt{a})^2 = a$.
The solution applies these rules step by step to simplify the given radical expression.
