Divide ( square root of 24)/8

Solution:

Step 1: Simplify the Numerator

Step 1.1: Express 24 as a Product of its Factors

• Decompose 24 into $2^2 \cdot 6$, which allows us to factor out a perfect square.

• The expression becomes $\frac{\sqrt{4 \cdot 6}}{8}$.

Step 1.1.1: Identify the Perfect Square in 24

• Recognize that 4 is a perfect square that can be factored out of 24.

• The expression now reads $\frac{\sqrt{2^2 \cdot 6}}{8}$.

Step 1.2: Extract Square Roots of Perfect Squares

• Take the square root of the perfect square $2^2$ out from under the radical.

• The expression simplifies to $\frac{2\sqrt{6}}{8}$.

Step 2: Reduce the Fraction

Step 2.1: Factor Out the Common Numerator

• Factor the numerator to isolate the common factor of 2.

• The expression is now $\frac{2(\sqrt{6})}{8}$.

Step 2.2: Eliminate Common Factors

• Identify and remove common factors between the numerator and the denominator.

Step 2.2.1: Factor the Common Factor from the Denominator

• Factor 2 out of 8, which gives us $\frac{2\sqrt{6}}{2 \cdot 4}$.

Step 2.2.2: Cancel the Common Factor

• Cancel out the common factor of 2.

• This reduces the expression to $\frac{\sqrt{6}}{4}$.

Step 2.2.3: Present the Simplified Expression

• The simplified expression remains $\frac{\sqrt{6}}{4}$.

Step 3: Provide the Result in Various Forms

• Exact Form: $\frac{\sqrt{6}}{4}$
• Decimal Form: Approximately $0.61237243 \ldots$

Knowledge Notes:

• Simplifying Square Roots: To simplify a square root, factor the number under the radical into its prime factors and identify pairs of prime factors, as pairs can be taken out from under the radical as a single number.

• Factoring Perfect Squares: A perfect square is a number that can be expressed as the square of an integer. Factoring out perfect squares from under a radical simplifies the expression.

• Reducing Fractions: To reduce fractions, find common factors in the numerator and denominator and divide them out. This simplification process is essential for achieving the simplest form of a fraction.

• Decimal Approximation: Some square roots do not result in whole numbers. In such cases, we can provide a decimal approximation. However, it is important to note that the decimal form of irrational numbers is non-terminating and non-repeating.

• Mathematical Notation: In mathematical expressions, the use of LaTeX formatting helps in clearly presenting numbers, variables, and operations, making complex expressions more readable.