Problem 12 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. \(\frac{3}{5} \sqrt{5}-\frac{1}{4} \sqrt{80}\)

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{2}{5} \sqrt{5}\).

Step 1: Simplify the Square Root

First, simplify the square root in the expression. The term \( \frac{1}{4} \sqrt{80} \) can be simplified by breaking down \( \sqrt{80} \) as \( \sqrt{16 \times 5} \). Therefore, \( \sqrt{80} = \sqrt{16} \times \sqrt{5} = 4 \sqrt{5} \).

Step 2: Substitute the Simplified Square Root Back

Replace \( \sqrt{80} \) with \( 4 \sqrt{5} \) in the expression. The expression becomes: \( \frac{3}{5} \sqrt{5} - \frac{1}{4} \times 4 \sqrt{5} \). Simplifying further, this results in \( \frac{3}{5} \sqrt{5} - 1 \sqrt{5} \).

Step 3: Factor Out the Common Term

Notice that both terms have \( \sqrt{5} \) as a common factor. Factor \( \sqrt{5} \) out: \( \left( \frac{3}{5} - 1 \right) \sqrt{5} \). Simplify the coefficients inside the parentheses.

Step 4: Simplify the Coefficients

Subtract the coefficients: \( \frac{3}{5} - 1 = \frac{3}{5} - \frac{5}{5} = -\frac{2}{5} \). Thus, the expression becomes \( -\frac{2}{5} \sqrt{5} \).

Key Concepts

Simplifying Expressions

When simplifying expressions, our goal is to make them as simple as possible. This often involves using properties like the distributive property or combining like terms. In our example, the given expression is

• \[\frac{3}{5} \sqrt{5} - \frac{1}{4} \sqrt{80}\]

To simplify this expression, we need to break down parts of it. Notice that both terms contain a form of the square root. Before combining these terms, it's useful to consider simplifying any elements like square roots or fractions, as these can often reduce the complexity of the expression. By simplifying and rewriting elements, we make further calculations easier and the entire expression more comprehensible.
By the end of this process, we aim for a reduced and cleaner form, which in our example is

• \[-\frac{2}{5} \sqrt{5}\].

Simplifying expressions is a crucial step that precedes other operations like factoring or solving equations, thereby making those processes more manageable.

Square Roots

Understanding square roots is key to many math problems, including simplifying expressions. The square root narrows down what number, when multiplied by itself, gives the original number. For example, the square root of 16 (\( \sqrt{16} \)) is 4 because 4 × 4 equals 16.
In our exercise, \( \sqrt{80} \) needs simplification. We do this by breaking it into components: \[\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}.\]
Recognizing that 16 is a perfect square helps us to express it as a simpler radical term, which is essential for further simplifications in mathematical expressions and equations.
Approaching square roots in this way not only helps with simplification but also aids in identifying equivalent expressions during factoring or when solving equations.

Factoring

Factoring is a powerful tool in mathematics that breaks down expressions into multiplication of simpler components. It is particularly handy when simplifying expressions or solving equations. In this context, noticing common factors, such as in our example involving \( \sqrt{5} \), is instrumental.
The exercise provided an expression ready for factoring:

• \[\frac{3}{5}\sqrt{5} - 1\sqrt{5}\],

Here, both terms have a common factor: \( \sqrt{5}. \
\)Factoring simplifies it by taking the common element outside the expression:

• \[\left( \frac{3}{5} - 1 \right) \sqrt{5}\],

makes calculations more direct. We achieve additional simplification by combining coefficients. As a result, such actions can drastically condense expressions and make them more workable for further mathematical operations. Factoring not only cleans up an expression but also sheds light on relationships between different terms within it.