Problem 35 - Algebra 1

Question

Simplify the expression. $$\sqrt{\frac{27}{36}}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{\sqrt{3}}{2}\)

Step 1: Square root of the numerator

The problem starts with finding the square root of 27. The square root of 27 is \(\sqrt{27} = 3\sqrt{3}\)

Step 2: Square root of the denominator

Next we find the square root of 36. The square root of 36 is \(\sqrt{36} = 6\)

Step 3: Simplify the fraction

Finally, we can simplify the fraction \(\frac{3\sqrt{3}}{6}\). By dividing both the numerator and the denominator by the greatest common divisor (3), the fraction simplifies to \(\frac{\sqrt{3}}{2}\)

Key Concepts

Square Root

When you encounter a square root in mathematics, such as \(\sqrt{x}\), you're looking for a number that, when multiplied by itself, gives you \(x\). Simplifying the square root of a number means breaking it down into its prime factors and pairing them to remove them from under the root. For instance, to simplify \(\sqrt{27}\), we recognize that \(27 = 3 \times 3 \times 3\) or \(3^3\). We extract pairs of primes, which gives us \(3\sqrt{3}\), since one pair of threes comes out of the radical as a single three.

Understanding this process is crucial for simplifying radical expressions and requires a strong grasp of multiplication and factors. Always look for perfect square factors within the radicand—the number under the square root—since these can be easily taken out to simplify the radical expression.

Simplify Fraction

A crucial skill in mathematics is the ability to simplify fractions. Simplifying a fraction means reducing it to its lowest terms, where the numerator and the denominator have no common factors other than 1. In our exercise, we have the fraction \(\frac{3\sqrt{3}}{6}\) resulting from taking the square roots of the numerator and denominator separately.

To simplify \(\frac{3\sqrt{3}}{6}\), observe that both the numerator and denominator can be divided by 3, the greatest common divisor of 3 and 6. Dividing both by 3 gives us \(\frac{\sqrt{3}}{2}\), which cannot be simplified any further since \(\sqrt{3}\) is an irrational number and does not have any common factors with 2. The key to simplifying fractions is to always look for the greatest common divisor between the numerator and the denominator and divide both by it to get a simpler, yet equivalent, fraction.

Greatest Common Divisor

The greatest common divisor (GCD), often referred to as the greatest common factor (GCF), of two or more integers is the largest positive integer that divides each of the numbers without leaving a remainder. Finding the GCD is an essential step in simplifying fractions, as it allows us to reduce fractions to their simplest form.

There are various methods to determine the GCD, such as listing out the factors of each number and finding the largest common one, or using the Euclidean algorithm for larger numbers. In the context of our exercise, the numbers 3 and 6 have common factors of 1 and 3, with 3 being the greatest. Thus, we use it to divide both the numerator and the denominator to simplify the fraction \(\frac{3\sqrt{3}}{6}\). It's important to be able to identify the GCD quickly and accurately to efficiently simplify fractions as part of algebraic manipulations and problem-solving.