Simplify square root of 18x^11y^7

Solution:

Step 1: Decompose the expression into perfect squares and remaining factors

Rewrite the expression $18x^{11}y^{7}$ as $(3x^{5}y^{3})^{2} \cdot 2xy$.

Step 1.1: Extract the square factor from 18

Factor 9 from 18 as $\sqrt{9 \cdot 2x^{11}y^{7}}$.

Step 1.2: Express 9 as a square of 3

Represent 9 as $3^{2}$, yielding $\sqrt{3^{2} \cdot 2x^{11}y^{7}}$.

Step 1.3: Separate the even power of x

Extract $x^{10}$ from $x^{11}$ as $\sqrt{3^{2} \cdot 2(x^{10}x)y^{7}}$.

Step 1.4: Rewrite $x^{10}$ as a square

Express $x^{10}$ as $(x^{5})^{2}$, resulting in $\sqrt{3^{2} \cdot 2((x^{5})^{2}x)y^{7}}$.

Step 1.5: Separate the even power of y

Factor $y^{6}$ from $y^{7}$ as $\sqrt{3^{2} \cdot 2((x^{5})^{2}x)(y^{6}y)}$.

Step 1.6: Rewrite $y^{6}$ as a square

Rewrite $y^{6}$ as $(y^{3})^{2}$, giving $\sqrt{3^{2} \cdot 2((x^{5})^{2}x)((y^{3})^{2}y)}$.

Step 1.7: Rearrange the x term

Rearrange to place the x term outside the perfect square as $\sqrt{3^{2} \cdot 2((x^{5})^{2})(y^{3})^{2}xy}$.

Step 1.8: Rearrange the 2 factor

Rearrange to place the 2 factor outside the perfect square as $\sqrt{(3^{2}((x^{5})^{2})(y^{3})^{2}) \cdot 2xy}$.

Step 1.9: Combine the perfect squares

Combine the perfect squares into a single term as $\sqrt{(3x^{5}y^{3})^{2} \cdot 2xy}$.

Step 1.10: Add parentheses around the non-square factors

Introduce parentheses to clearly separate the non-square factors as $\sqrt{(3x^{5}y^{3})^{2} \cdot (2xy)}$.

Step 1.11: Ensure the expression is properly parenthesized

Verify that the expression is correctly parenthesized as $\sqrt{(3x^{5}y^{3})^{2} \cdot (2xy)}$.

Step 2: Simplify the radical expression

Extract the perfect square terms from under the radical to obtain $3x^{5}y^{3}\sqrt{2xy}$.

Knowledge Notes:

To simplify a square root involving variables and coefficients, we follow these steps:

1. Factor the coefficient into a product of perfect squares and other factors.

2. Rewrite each variable with an even exponent as the square of a variable with half that exponent.

3. Separate the perfect squares from the non-square terms.

4. Extract the square root of the perfect squares, which can be done without a radical since they are perfect squares.

5. Leave the non-square terms under the radical.

In this problem, we are given the expression $\sqrt{18x^{11}y^{7}}$. We need to simplify it by finding perfect square factors and extracting them from under the radical. We use the properties of exponents and the fact that the square root of a perfect square is just the base of the exponent. We also use the associative property of multiplication to group terms under the radical in a way that reveals perfect squares. The final result is a simplified expression with some terms outside the radical (those that were perfect squares) and the remaining terms inside the radical.