Problem 48 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{\frac{5}{6}} $$

Step-by-Step Solution

Answer

The most simplified form of the given expression is \( \frac{\sqrt{5}}{\sqrt{2} \times \sqrt{3}} \).

Step 1: Understanding the Problem

The expression given is \( \sqrt{\frac{5}{6}} \), a square root of a fraction. We know that a square root of a fraction can be re-written as a fraction of the square roots of the numerator and the denominator respectively.

Step 2: Simplification

Therefore, the expression can be re-written as \( \frac{\sqrt{5}}{\sqrt{6}} \). The square root of 5 cannot be simplified any further. However, the square root of 6 can be simplified by factoring 6 into 2 and 3, each of which are prime numbers. Therefore, the square root of 6 simplifies to \( \sqrt{2} \times \sqrt{3} \), which cannot be simplified any further as both are prime numbers.

Step 3: Final Simplification

The final simplified expression is \( \frac{\sqrt{5}}{\sqrt{2} \times \sqrt{3}} \). There is no numerical solution to this exercise, as the square roots of neither 2, 3, nor 5 can be represented as exact decimals or fractions. Therefore, the most simplified form of the expression is \( \frac{\sqrt{5}}{\sqrt{2} \times \sqrt{3}} \).

Key Concepts

Square Roots

Square roots are fundamental in mathematics, often appearing in equations and formulas. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because \( 4 \times 4 = 16 \). When you're simplifying expressions involving square roots, it helps to understand their properties:
- A square root can be expressed as a power; specifically, square root of a number \( x \) is \( x^{1/2} \).
- The square root of a product of numbers is the product of their square roots, i.e., \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).In the given problem, \( \sqrt{\frac{5}{6}} \) can be broken down using the property of square roots over fractions, \( \sqrt{\frac{5}{6}} = \frac{\sqrt{5}}{\sqrt{6}} \).
This step captures the essence of simplifying the problems involving square roots.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics, helping in making calculations easier and expressions clearer. To simplify a fraction \( \frac{a}{b} \), we divide both the numerator \( a \) and the denominator \( b \) by their greatest common divisor (GCD). However, when the numerator or denominator contains a square root, the process is slightly different.
In the expression from the exercise, \( \frac{\sqrt{5}}{\sqrt{6}} \), simplifying requires us to look at the square roots individually.For square roots, simplify by factoring the number inside the root to its prime factors:
- 5 is already a prime number, so \( \sqrt{5} \) remains unchanged.
- 6 can be broken down into \( 2 \times 3 \), therefore \( \sqrt{6} = \sqrt{2 \times 3} \).
While neither the numerator nor the denominator can be reduced further in terms of numerical square root values, understanding this process is crucial for handling more complex expressions.

Prime Factorization

Prime factorization is a technique where a number is broken down into the product of prime numbers. This is particularly useful in simplifying square roots because it helps identify if any factors could pair up to come out of the square root.For example, the number 6 can be expressed in terms of its prime factors as \( 6 = 2 \times 3 \).
Each of these factors can be rewritten within a square root; thus, \( \sqrt{6} \) becomes \( \sqrt{2} \times \sqrt{3} \).
Prime factorization simplifies identifying simplifications possible when dealing with square roots.
In the fraction, \( \frac{\sqrt{5}}{\sqrt{6}} \), prime factorization helps illustrate that the denominator in this case, \( \sqrt{6} = \sqrt{2 \times 3} \), is in its simplest term, as neither 2 nor 3 pair up to create a whole number outside the square root.
Using prime factorization in solving problems enhances your understanding and ensures expressions are as simplified as possible.