Problem 325 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{\sqrt{80}}{\sqrt{125}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{4}{5}\).

Step 1: Simplify both square roots separately

First, simplify \(\root{80}\) and \(\root{125}\) individually. \(\root{80} = \root{(16 \times 5)} = \root{16} \times \root{5} = 4\root{5}\) and \(\root{125} = \root{(25 \times 5)} = \root{25} \times \root{5} = 5\root{5}\).

Step 2: Substitute the simplified expressions

Replace the square roots in the original fraction with their simplified forms. This gives \(\frac{4\root{5}}{5\root{5}}\).

Step 3: Simplify the fraction

Divide both the numerator and the denominator by \(\root{5}\). This leaves \(\frac{4}{5}\).

Key Concepts

Simplifying square roots

Simplifying square roots means finding the simplest expression possible for a given square root. This often involves breaking down the number inside the square root into its prime factors.
For example, to simplify \(\root{80}\), we find the prime factorization of 80: \(80 = 2^4 \times 5\). Since \(\root{(16 \times 5)} = \root{16} \times \root{5} = 4\root{5}\), we simplify \(\root{80}\) to 4\(\root{5}\).
Similarly, for \(\root{125}\), we break it down as \(\root{(25 \times 5)} = \root{25} \times \root{5} = 5\root{5}\).
The key is to look for perfect square factors within the original number, making the square root simpler. Don't forget, practice makes perfect when identifying these factors.

Fraction simplification

Fraction simplification involves reducing fractions to their simplest form. To simplify fractions with square roots, we first simplify the square roots themselves.
Once we have the simplified square roots, like \(4\root{5}\) and \(5\root{5}\) from earlier, we replace these into the fraction. This gives us \(\frac{4\root{5}}{5\root{5}}\).
Next, we look for common factors in the numerator and the denominator. Here, \(\root{5}\) is a common factor. Dividing both the numerator and the denominator by \(\root{5}\), we are left with \(\frac{4}{5}\).
Simplifying fractions can help in many math problems, making them easier to solve or understand. Always look for common factors to reduce your fraction smoothly.

Radical simplification

Radical simplification is the process of making radical expressions simpler and more manageable. Radicals can be square roots, cube roots, or other roots.
In our example, we simplified \( \root{80}\) to \(4\root{5} \) and \( \root{125}\) to \(5\root{5} \). By identifying perfect squares within the radicals, we could separate and reduce them.
Once substituted into the fraction, simplifying \( \frac{4\root{5}}{5\root{5}} \) further involves removing common radical factors. Here, \(\root{5}\) cancels out, resulting in \( \frac{4}{5} \).
The steps we use for radical simplification are useful across many different problems. Always look for factors to break down the radicals and simplify step-by-step to reach the simplest form.