Question

Addition and Subtraction of Radicals. Combine as indicated and simplify. $$3 \sqrt{\frac{5}{4}}+2 \sqrt{45}$$

Short answer

\(\frac{15}{4} + 6\sqrt{5}\)

Step-by-step solution

- Simplify the Radicals

Start by simplifying each radical. The first term has a rational number inside the radical, which can be simplified by taking the square root of the numerator and the denominator separately: \(3 \sqrt{\frac{5}{4}} = 3 \sqrt{\frac{\sqrt{25}}{\sqrt{16}}} = 3 \frac{\sqrt{25}}{\sqrt{16}} = 3 \frac{5}{4} = \frac{15}{4}\). The second term can be simplified by recognizing that \(45\) is \(9\times5\) and \(9\) is a perfect square: \(2 \sqrt{45} = 2 \sqrt{9 \times 5} = 2 \sqrt{9} \sqrt{5} = 2 \times 3 \sqrt{5} = 6\sqrt{5}\).

- Combine Like Terms

Since the radicals have been simplified, identify and combine like terms. In this case, after simplification, there are no like terms to combine, because we have a rational number and a radical expression. Thus, the simplified expression becomes \(\frac{15}{4} + 6\sqrt{5}\). Since there are no like terms, this is the final simplified form.

Key concepts

Radical Expressions

A radical expression is an expression that includes a square root, cube root, or higher roots. The square root symbol (√) is known as a radical sign, and the term underneath the radical sign is called the radicand. Simplifying radical expressions requires some knowledge of integer factorization, perfect squares, and the properties of radicals.

When simplifying radical expressions like the ones in the exercise, we look for factors that are perfect squares. For example, when the radicand is a fraction as in \(3 \sqrt{\frac{5}{4}}\), we can take the square root of the numerator and denominator separately if they are perfect squares. The process of simplifying involves reducing the expression to its simplest form by performing operations both inside and outside the radical.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. The square root of a perfect square results in an integer, such as \(\sqrt{9} = 3\). However, if the number is not a perfect square, the square root can be an irrational number, and it might not be possible to simplify it without approximating.

During simplification, knowing your perfect squares (such as 1, 4, 9, 16, 25, and so on) helps greatly. Simplified square roots do not contain fractions inside the radical and have no square roots in the denominator, so rationalizing denominators may be necessary. For instance, \(\sqrt{\frac{25}{16}}\) simplifies by taking the square root of the top and bottom, resulting in \(\frac{\sqrt{25}}{\sqrt{16}} = \frac{5}{4}\), just as demonstrated in the step-by-step solution.

Addition of Radicals

The addition of radicals involves combining radical expressions that have the same radicand and index, similar to combining like terms in algebraic expressions. If the radicals differ, we must first simplify them to see if they can be combined.

Addition is only possible when the radicands and their indices are identical after simplification. As seen in the provided example, the addition \(\frac{15}{4} + 6\sqrt{5}\) cannot be further simplified by addition, since one term is a rational number and the other is a radical expression.

Exercise Improvement Advice

It is important to remember that impossible addition is often a point of confusion. When instructing on the topic, reinforce the concept that only like radicals can be added - the same concept as combining only like terms in algebra. If students try to combine terms that are not similar radicals, revisiting the fundamental properties of radicals and practicing more problems where simplification might lead to like radicals can help solidify their understanding and improve their skills.