Problem 60 - Elementary and Intermediate Algebra

Question

Simplify by combining like radicals. $$ \sqrt{20}+\sqrt{125}-\sqrt{80} $$

Step-by-Step Solution

Answer

The simplified expression is \(3\sqrt{5}\).

Step 1: Simplify Individual Radicals

First, simplify each of the square roots by identifying perfect square factors. \(\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}\)\(\sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5}\)\(\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}\).

Step 2: Combine Like Terms

Now, combine the terms as they all have \(\sqrt{5}\) as a common factor:\(2\sqrt{5} + 5\sqrt{5} - 4\sqrt{5}\)

Step 3: Perform Addition and Subtraction

Add and subtract the coefficients of \(\sqrt{5}\):\((2 + 5 - 4)\sqrt{5} = 3\sqrt{5}\).

Key Concepts

Combining Like Terms

When simplifying expressions with square roots, the process of combining like terms becomes essential. Just as with algebraic terms, where we add or subtract coefficients of similar variables, combining like radicals involves arithmetic on coefficients of identical square root terms. In the original exercise, each term simplifies to a variant of \(\sqrt{5}\).

• \(2\sqrt{5}\) from \(\sqrt{20}\)
• \(5\sqrt{5}\) from \(\sqrt{125}\)
• \(4\sqrt{5}\) from \(\sqrt{80}\)

These expressions are like terms because they share the common radical part, \(\sqrt{5}\). This similarity in the radical component allows us to perform simple arithmetic, adding and subtracting their coefficients just like you would with variables. This simplifies the expression to \(3\sqrt{5}\). Through combining like terms, we ensure the functionals of previously complex square roots are considerably easier to work with.

Perfect Square Factors

Identifying perfect square factors is crucial to simplifying radical expressions. A perfect square factor is a number made by squaring an integer. For example, 4, 9, 16, and 25 are perfect squares because they equal \(2^2\), \(3^2\), \(4^2\), and \(5^2\) respectively.
Breaking down a number into a product of perfect square factors helps simplify square roots. In our example:

• For \(\sqrt{20}\), the factor is 4, giving \(\sqrt{4 \times 5} = 2\sqrt{5}\).
• In \(\sqrt{125}\), 25 is the factor, leading to \(\sqrt{25 \times 5} = 5\sqrt{5}\).
• Lastly, \(\sqrt{80}\) uses 16 as the factor, simplifying to \(\sqrt{16 \times 5} = 4\sqrt{5}\)

Utilizing perfect square factors simplifies complex radical expressions by reducing them to more manageable forms, in turn making operations such as addition and subtraction far easier to perform.

Square Roots

Square roots play a vital role in mathematics, offering solutions to equations involving second-degree variables. The square root essentially asks, "what number, when multiplied by itself, results in this particular value?" For instance, \(\sqrt{9} = 3\) because \(3^2 = 9\). Understanding how square roots work and how to simplify them is key to handling radical expressions.

To simplify a radical, you first identify any perfect square factors within the number under the root. Decomposition allows you to "extract" these perfect square factors out from under the radical sign, simplifying the task of working with or combining the resulting expressions. When factors are not squares, they remain under the radical sign, resulting in expressions like \(2\sqrt{5}\). Mastering square roots involves practicing this simplification process and recognizing when terms can be combined for easier computation.