Simplify 3 square root of -27x^6

Solution:

Step 1: Express -27x^6 in a different form

Start by rewriting $-27x^6$ as $(3x^3)^2 \cdot -3$.

Step 1.1: Extract the square factor from -27

Factor out 9 from $-27$ to get $3\sqrt{9(-3)x^6}$.

Step 1.2: Represent 9 as a square of 3

Express 9 as $3^2$ to get $3\sqrt{3^2 \cdot -3x^6}$.

Step 1.3: Express x^6 as a square

Rewrite $x^6$ as $(x^3)^2$ to obtain $3\sqrt{3^2 \cdot -3(x^3)^2}$.

Step 1.4: Rearrange the -3 term

Move the $-3$ term to get $3\sqrt{3^2(x^3)^2 \cdot -3}$.

Step 1.5: Combine the squares

Combine $3^2$ and $(x^3)^2$ to form $(3x^3)^2$ resulting in $3\sqrt{(3x^3)^2 \cdot -3}$.

Step 2: Simplify the radical expression

Extract terms from under the radical to get $3(3x^3\sqrt{-3})$.

Step 3: Factor out -1 from -3

Rewrite $-3$ as $-1(3)$ to obtain $3(3x^3\sqrt{-1(3)})$.

Step 4: Separate the square roots

Separate the square roots of $-1$ and $3$ to get $3(3x^3(\sqrt{-1} \cdot \sqrt{3}))$.

Step 5: Replace the square root of -1 with i

Substitute $\sqrt{-1}$ with $i$ to have $3(3x^3(i \cdot \sqrt{3}))$.

Step 6: Final multiplication

Multiply 3 by 3 to get the final answer $9x^3i\sqrt{3}$.

Knowledge Notes:

The problem involves simplifying a radical expression with a negative radicand and an even power of a variable. Here are the relevant knowledge points:

1. Radicals and Powers: The square root of a variable raised to an even power can be simplified by taking the square root of the base raised to half the power.

2. Negative Radicands: The square root of a negative number involves the imaginary unit $i$, where $i^2 = -1$.

3. Factoring: Factoring involves rewriting an expression as a product of its factors. In this case, -27 is factored into 9 and -3, and 9 is further expressed as $3^2$.

4. Simplifying Radical Expressions: When a radical contains a perfect square, that part can be taken out of the radical, simplifying the expression.

5. Imaginary Numbers: The square root of -1 is represented by the imaginary unit $i$. This is used when dealing with the square roots of negative numbers.

6. Multiplication of Radicals: When multiplying radicals, you can multiply the numbers outside the radicals together and the numbers inside the radicals together.

7. Combining Like Terms: After simplifying the expression inside the radical, like terms are combined, such as multiplying constants outside the radical.

By understanding these concepts, one can simplify complex radical expressions, even when they involve imaginary numbers.