Simplify 9 square root of 32x^12y^12

Solution:

Step 1: Express the expression inside the square root in a different form

Express $32x^{12}{y}^{12}$ as $(4x^6{y}^6{)}^2\cdot 2$.

Step 1.1: Separate the perfect square from the product

Identify the perfect square factor in $32$, which is $16$. Thus, we have $9\sqrt{16\cdot 2x^{12}{y}^{12}}$.

Step 1.2: Represent $16$ as a square of $4$

Recognize that $16$ is the square of $4$, so write it as $9\sqrt{4^2\cdot 2x^{12}{y}^{12}}$.

Step 1.3: Rewrite $x^{12}$ as a square

Notice that $x^{12}$ is a perfect square and can be written as $(x^6{)}^2$. Now we have $9\sqrt{4^2\cdot 2(x^6{)}^2{y}^{12}}$.

Step 1.4: Rewrite $y^{12}$ as a square

Similarly, $y^{12}$ is a perfect square and can be written as $(y^6{)}^2$. The expression becomes $9\sqrt{4^2\cdot 2(x^6{)}^2(y^6{)}^2}$.

Step 1.5: Rearrange the terms under the radical

Reorder the terms under the square root to group the perfect squares together: $9\sqrt{4^2(x^6{)}^2(y^6{)}^2\cdot 2}$.

Step 1.6: Combine the perfect squares under the radical

Combine the perfect squares into a single term: $9\sqrt{(4x^6{y}^6{)}^2\cdot 2}$.

Step 2: Simplify the square root by extracting the perfect square

Extract the perfect square from under the radical: $9(4x^6{y}^6)\sqrt{2}$.

Step 3: Multiply the extracted term by the coefficient outside the radical

Multiply the coefficient $9$ by the extracted term $4$: $36x^6{y}^6\sqrt{2}$.

Knowledge Notes:

The problem involves simplifying a radical expression with a numerical coefficient. The key steps in simplifying such expressions include:

1. Identifying and factoring out perfect squares from the expression under the radical.

2. Using the property that $\sqrt{a^2}=a$ to simplify the expression, where $a$ is a non-negative real number.

3. Rearranging terms to group perfect squares together, which makes it easier to extract them from under the square root.

4. Multiplying the extracted terms by any coefficients outside the radical to obtain the final simplified expression.

In this problem, we used the fact that $32=16\cdot 2$ and $16$ is a perfect square, $16=4^2$. We also used the property that any even power of a variable can be expressed as a square, for example, $x^{12}=(x^6{)}^2$ and $y^{12}=(y^6{)}^2$. By recognizing these perfect squares, we were able to simplify the radical expression.