Simplify ( square root of 16x^5y^12)/( square root of 36xy^2)

Solution:

Step 1:

Merge $\sqrt{16 x^{5} y^{12}}$ with $\sqrt{36 x y^{2}}$ to form a single square root expression: $\sqrt{\frac{16 x^{5} y^{12}}{36 x y^{2}}}$.

Step 2:

Simplify the fraction $\frac{16 x^{5} y^{12}}{36 x y^{2}}$ by removing common factors.

Step 2.1:

Extract the factor of $4$ from the numerator $16 x^{5} y^{12}$: $\sqrt{\frac{4(4 x^{5} y^{12})}{36 x y^{2}}}$.

Step 2.2:

Extract the factor of $4$ from the denominator $36 x y^{2}$: $\sqrt{\frac{4(4 x^{5} y^{12})}{4(9 x y^{2})}}$.

Step 2.3:

Eliminate the common factor of $4$: $\sqrt{\frac{\cancel{4}(4 x^{5} y^{12})}{\cancel{4}(9 x y^{2})}}$.

Step 2.4:

Reformulate the expression: $\sqrt{\frac{4 x^{5} y^{12}}{9 x y^{2}}}$.

Step 3:

Remove the common $x$ powers from $x^{5}$ and $x$.

Step 3.1:

Isolate $x$ from the numerator $4 x^{5} y^{12}$: $\sqrt{\frac{x(4 x^{4} y^{12})}{9 x y^{2}}}$.

Step 3.2:

Discard the common $x$ factors.

Step 3.2.1:

Isolate $x$ from the denominator $9 x y^{2}$: $\sqrt{\frac{x(4 x^{4} y^{12})}{x(9 y^{2})}}$.

Step 3.2.2:

Eliminate the common $x$ factor: $\sqrt{\frac{\cancel{x}(4 x^{4} y^{12})}{\cancel{x}(9 y^{2})}}$.

Step 3.2.3:

Rephrase the expression: $\sqrt{\frac{4 x^{4} y^{12}}{9 y^{2}}}$.

Step 4:

Reduce the $y$ powers by canceling out common factors from $y^{12}$ and $y^{2}$.

Step 4.1:

Extract $y^{2}$ from the numerator $4 x^{4} y^{12}$: $\sqrt{\frac{y^{2}(4 x^{4} y^{10})}{9 y^{2}}}$.

Step 4.2:

Remove the common $y^{2}$ factors.

Step 4.2.1:

Extract $y^{2}$ from the denominator $9 y^{2}$: $\sqrt{\frac{y^{2}(4 x^{4} y^{10})}{y^{2} \cdot 9}}$.

Step 4.2.2:

Cancel the common $y^{2}$ factor: $\sqrt{\frac{\cancel{y^{2}}(4 x^{4} y^{10})}{\cancel{y^{2}} \cdot 9}}$.

Step 4.2.3:

Restate the expression: $\sqrt{\frac{4 x^{4} y^{10}}{9}}$.

Step 5:

Represent $4 x^{4} y^{10}$ as $(2 x^{2} y^{5})^{2}$: $\sqrt{\frac{(2 x^{2} y^{5})^{2}}{9}}$.

Step 6:

Express $9$ as $3^{2}$: $\sqrt{\frac{(2 x^{2} y^{5})^{2}}{3^{2}}}$.

Step 7:

Rewrite $\frac{(2 x^{2} y^{5})^{2}}{3^{2}}$ as $(\frac{2 x^{2} y^{5}}{3})^{2}$: $\sqrt{(\frac{2 x^{2} y^{5}}{3})^{2}}$.

Step 8:

Extract terms from under the radical, assuming all are positive real numbers: $\frac{2 x^{2} y^{5}}{3}$.

Knowledge Notes:

The problem involves simplifying a radical expression containing a fraction. The steps taken to simplify the expression include:

1. Combining the radicals into a single radical expression.

2. Factoring out common terms from the numerator and the denominator.

3. Canceling out common factors to simplify the fraction inside the radical.

4. Recognizing perfect squares and simplifying them to remove the radical.

Key concepts used in this problem include:

• Radical Simplification: The process of simplifying expressions under a radical sign, often by factoring out perfect squares or cubes, depending on the index of the radical.

• Fraction Reduction: The process of simplifying a fraction by canceling out common factors from the numerator and the denominator.

• Algebraic Manipulation: The use of algebraic rules to rewrite expressions in a simpler or more convenient form.

• Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of a perfect square is always an integer.

The final result is obtained by applying these concepts systematically to simplify the given expression.