Divide ( square root of 6)÷( square root of 2)

Solution:

Step 1:

Express the division as a ratio using a single fraction notation: $\frac{\sqrt{6}}{\sqrt{2}}$.

Step 2:

Merge the square roots in the numerator and the denominator into one square root: $\sqrt{\frac{6}{2}}$.

Step 3:

Simplify the fraction inside the radical by dividing $6$ by $2$: $\sqrt{3}$.

Step 4:

Present the solution in its various acceptable forms. In its exact form, the solution is $\sqrt{3}$. If converted to decimal, the solution is approximately $1.73205080 \ldots$.

Knowledge Notes:

1. Rationalizing the Denominator: When dividing square roots, it is common to combine them into a single square root to simplify the expression. This process is akin to rationalizing the denominator when dealing with fractions that contain radicals.

2. Simplifying Square Roots: To simplify a square root of a fraction, you can take the square root of the numerator and the denominator separately. This is based on the property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

3. Exact vs. Decimal Form: The exact form of a radical expression is often preferred in mathematics because it maintains precision. However, for practical applications, converting to a decimal form may be necessary. Keep in mind that the decimal representation of irrational numbers is approximate and often truncated or rounded.

4. Irrational Numbers: The square root of a non-perfect square (like 3) is an irrational number. It cannot be expressed as a simple fraction and has an infinite, non-repeating decimal expansion.

5. Notation: In mathematical notation, the square root symbol $\sqrt{}$ is used to denote the principal square root of a number. The principal square root of a positive number is always positive, and for a non-negative number $a$, it is the positive number that, when multiplied by itself, gives $a$.