Problem

Divide ( square root of 24)/8

The given problem is a mathematical expression that needs to be simplified. It involves the operation of division where the dividend is the square root of 24, and the divisor is the number 8. The question is asking to perform this division operation to reduce the expression to its simplest form.

248

Answer

Expert–verified

Solution:

Step 1: Simplify the Numerator

Step 1.1: Express 24 as a Product of its Factors

  • Decompose 24 into 226, which allows us to factor out a perfect square.

  • The expression becomes 468.

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 2268.

Step 1.2: Extract Square Roots of Perfect Squares

  • Take the square root of the perfect square 22 out from under the radical.

  • The expression simplifies to 268.

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 2(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 2624.
Step 2.2.2: Cancel the Common Factor
  • Cancel out the common factor of 2.

  • This reduces the expression to 64.

Step 2.2.3: Present the Simplified Expression
  • The simplified expression remains 64.

Step 3: Provide the Result in Various Forms

  • Exact Form: 64
  • Decimal Form: Approximately 0.61237243

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.

link_gpt