Problem 82 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. \(\frac{1}{5} \sqrt{500}\)

Step-by-Step Solution

Answer

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

Step 1: Simplify the radicand

The first step is to simplify the radicand (the number under the square root). As the 500 is a perfect square of 100 and 5, it can be rewritten as \(\sqrt{100 × 5}\) or \(\sqrt{100} × \sqrt{5}\)

Step 2: Calculate the square root

The square root of 100 can be calculated while \(\sqrt{5} \) cannot be simplified further. Hence, \(\sqrt{100 × 5} = 10 \sqrt{5}\)

Step 3: Multiply the fractions

Now, multiply the fraction \(\frac{1}{5}\) with \(10 \sqrt{5}\). The multiplication gives a result of \(2 \sqrt{5}\)

Key Concepts

Square Root Simplification

Understanding how to simplify square root expressions is a fundamental skill in algebra. A square root, denoted by the square root symbol \( \sqrt{\ } \) , represents a number which, when multiplied by itself, gives the original number or the radicand. The process of square root simplification involves breaking down the radicand into its prime factors and identifying pairs of identical factors.

For instance, simplifying \( \sqrt{500} \) entails recognizing that 500 can be factored into \( 100 \times 5 \) . Since 100 is a perfect square (because \( 10 \times 10 = 100 \) ), it can be taken out from under the square root, leaving us with \( 10 \sqrt{5} \). It's important to look for the largest perfect square factor, as this reduces the radicand most effectively, making simplification easier and your expression neater.

Radicand Simplification

The radicand simplification process is crucial in streamlining the expression under a radical. This not only involves finding the largest perfect square, as seen with square root simplification but also applies to higher-degree roots. Simplifying the radicand can also mean breaking it down into its simplest radical form when a perfect square is not present.

For example, the radicand in \( \sqrt{500} \) can be simplified by realizing that 500 equals \( 100 \times 5 \) . Here \( 100 \) is the largest perfect square factor and \( 5 \) is left inside the square root, since it’s not a perfect square. This is illustrated in Step 1 of our given solution and is essential in finding the most simplified form possible for any radical expression.

Multiplying Fractions with Radicals

When dealing with radicals, it's not uncommon to encounter them as part of a fraction. Multiplying fractions with radicals follows the same principles as normal fraction multiplication — we multiply the numerators together and the denominators together. However, when a radical is involved, we treat it just like any other term.

If the radical can be simplified before multiplication, as shown in our solution's Step 2, it should be done to make calculations easier. In Step 3, the fraction \( \frac{1}{5} \) is multiplied by the simplified radical form \( 10 \sqrt{5} \) , resulting in \( 2 \sqrt{5} \) . It is essential to simplify radicals first to ensure that the multiplication yields the simplest form of expression. This also minimizes errors and provides a clearer understanding of the number or expression you're working with.