Problem 317 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{\sqrt{27}}{6} $$

Step-by-Step Solution

Answer

\frac{\root 3}{2}

Step 1: Simplify the square root

The first step is to simplify the square root in the numerator. The square root of 27 can be written as \(\root 27 = \root 3^3 = 3 \root 3\), since \(\root 27 = 3 \root 3\).

Step 2: Rewrite the fraction

Now, rewrite the fraction with the simplified numerator: \(\frac{3\root 3}{6}\).

Step 3: Simplify the fraction

Next, simplify the fraction \(\frac{3\root 3}{6}\). The 3 in the numerator and the 6 in the denominator have a common factor of 3. Divide both the numerator and the denominator by 3, resulting in: \(\frac{\root 3}{2}\).

Key Concepts

Square Roots

In mathematics, a square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9. The symbol for square root is \(\root x\), where x is the number. It's important to understand that some numbers have perfect square roots, like 16 (which is 4), while others do not, like 27. For the square root of 27, we can break it down as \(\root 27 = \root 3^3 = 3 \root 3\). This is because 27 can be factored into 3 times 3 times 3. Understanding how to simplify square roots is crucial when dealing with radicals, especially in fractions.

Fractions

Fractions represent a part of a whole. They consist of a numerator and a denominator, with the numerator being the number above the fraction line and the denominator being the number below it. In our example, we have \(\frac{\root 27}{6}\). The fraction tells us that the square root of 27 is being divided by 6. To simplify fractions, we often need to find common factors for the numerator and the denominator. A simplified fraction has the smallest possible integers. Understanding the basics of fractions plays a vital role in simplifying expressions because we need to manipulate both the numerator and the denominator correctly.

Simplification

Simplification is the process of making an expression easier to understand or solve. For our example, we started simplifying by breaking down the square root of 27 into \(\root 27 = 3 \root 3\). Next, we rewrote the fraction as \(\frac{3\root 3}{6}\). To further simplify, we identified that both 3 and 6 have a common factor, which is 3. Dividing both the numerator and the denominator by this common factor reduces the fraction to \(\frac{\root 3}{2}\). When simplifying, always look for common factors, break down complex numbers into simpler parts, and ensure the result is in its simplest form for easier interpretation.