Simplify square root of 54x^21

Solution:

Step 1: Decompose $54x^{21}$ into prime factors and perfect squares.

1.1 Extract $9$ from $54$, resulting in $\sqrt{9 \cdot 6x^{21}}$.

1.2 Recognize that $9$ is $3^2$, giving us $\sqrt{3^2 \cdot 6x^{21}}$.

1.3 Separate $x^{20}$ from $x^{21}$, yielding $\sqrt{3^2 \cdot 6(x^{20}x)}$.

1.4 Express $x^{20}$ as $(x^{10})^2$, leading to $\sqrt{3^2 \cdot 6((x^{10})^2x)}$.

1.5 Rearrange to place $6$ after the perfect square, obtaining $\sqrt{3^2 (x^{10})^2 \cdot 6x}$.

1.6 Combine $3^2$ and $(x^{10})^2$ into a single squared term, resulting in $\sqrt{(3x^{10})^2 \cdot 6x}$.

1.7 Enclose $6x$ in parentheses to emphasize the separate term, arriving at $\sqrt{(3x^{10})^2 \cdot (6x)}$.

Step 2: Simplify the radical expression.

Extract the perfect square $(3x^{10})^2$ from under the radical, simplifying to $3x^{10}\sqrt{6x}$.

Knowledge Notes:

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

1. Factor the coefficient into its prime factors and identify any perfect squares.

2. Rewrite the expression by separating the perfect squares from the non-perfect squares.

3. For the variable part, factor out the highest even power of the variable that is less than or equal to the current power.

4. Express the even power of the variable as a square of its half-power (e.g., $x^{20}$ as $(x^{10})^2$).

5. Simplify the square root by taking the square root of the perfect squares and leaving the rest under the radical.

6. Combine the terms outside the radical to get the simplified expression.

In this case, we used the fact that $54$ can be factored into $9 \cdot 6$, where $9$ is a perfect square, and $x^{21}$ can be separated into $x^{20} \cdot x$, where $x^{20}$ is a perfect square. The square root of a perfect square is simply the number or variable without the square, which is why we can take $3x^{10}$ out of the square root.