Simplify square root of 48x^14

Solution:

Simplification Process:

Step 1:
Express $48x^{14}$ as a product of squares and other factors.
Step 1.1:
Identify the square factor within 48, which is 16, and represent it as such: $\sqrt{16 \cdot 3x^{14}}$.
Step 1.2:
Recognize that 16 is the square of 4, and write it as $4^2$: $\sqrt{4^2 \cdot 3x^{14}}$.
Step 1.3:
Notice that $x^{14}$ is the square of $x^7$, and represent it accordingly: $\sqrt{4^2 \cdot 3(x^7)^2}$.
Step 1.4:
Rearrange the terms to group the squares together: $\sqrt{(4x^7)^2 \cdot 3}$.
Step 1.5:
Combine the squares under a single radical: $\sqrt{(4x^7)^2 \cdot 3}$.

Step 2:
Extract the square terms from under the radical to simplify: $4x^7\sqrt{3}$.

Knowledge Notes:

To simplify the square root of an algebraic expression, one can follow these steps:

1. Factorization: Break down the number and variable parts of the expression into factors, especially looking for perfect squares.

2. Rewrite as Squares: Express the factors as squares where possible, since the square root of a square is a simpler expression.

3. Radical Simplification: Simplify the expression under the square root by taking out the square terms, leaving the non-square terms under the radical.

In this case, the number 48 is factored into 16 (which is a perfect square, $4^2$) and 3. The variable part $x^{14}$ is rewritten as $(x^7)^2$, which is also a perfect square. By extracting the square terms from under the radical, the expression is simplified to $4x^7\sqrt{3}$.

Relevant mathematical properties used in this process include:

• The square root of a product is equal to the product of the square roots of the factors: $\sqrt{ab} = \sqrt{a}\sqrt{b}$.

• The square root of a square is the base of the square: $\sqrt{a^2} = a$.

• The exponent rule $(x^m)^n = x^{mn}$, which in this case simplifies $x^{14}$ as $(x^7)^2$.