Simplify square root of 300x^10

Solution:

Step 1:

Express $300x^{10}$ as a product of a perfect square and another term: $(10x^5)^2 \cdot 3$.

Step 1.1:

Extract $100$ from $300$: $\sqrt{100 \cdot 3x^{10}}$.

Step 1.2:

Represent $100$ as a square of $10$: $\sqrt{(10)^2 \cdot 3x^{10}}$.

Step 1.3:

Express $x^{10}$ as a square of $x^5$: $\sqrt{(10)^2 \cdot 3(x^5)^2}$.

Step 1.4:

Rearrange the terms: $\sqrt{(10)^2 (x^5)^2 \cdot 3}$.

Step 1.5:

Combine the squares into a single square term: $\sqrt{(10x^5)^2 \cdot 3}$.

Step 2:

Extract the square root of the perfect square: $10x^5\sqrt{3}$.

Knowledge Notes:

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

1. Factorization: Break down the expression inside the square root into factors that can help identify perfect squares.

2. Perfect Squares: Recognize perfect squares that can be taken out of the square root. A perfect square is a number that can be expressed as the square of an integer or a variable raised to an even power.

3. Extraction: Extract the square root of perfect squares by taking them outside the square root symbol. The square root of a perfect square is the number that, when multiplied by itself, gives the original number.

4. Simplification: Simplify the expression by multiplying the terms outside the square root and leaving the rest inside the square root if they are not perfect squares.

5. Algebraic Manipulation: Use algebraic rules to rearrange and combine like terms for a simplified expression.

In the given problem, $300x^{10}$ can be broken down into a perfect square, $(10x^5)^2$, and a non-perfect square, $3$. The perfect square is taken out of the square root, while the non-perfect square remains under the radical.