Question

Simplify the expression. $$4 \sqrt{5}+\sqrt{125}+\sqrt{45}$$

Short answer

The simplified expression is \(12\sqrt{5}\).

Step-by-step solution

Identify and simplify square roots

Start by simplifying the square roots that can be factored. Both \(\sqrt{125}\) and \(\sqrt{45}\) can be simplified by factoring out squares from under the radical. We have \(\sqrt{125} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5}\) and \(\sqrt{45} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}\).

Substitute simplified roots

Next, substitute these simplified square roots back into the original expression, which gives us \(4\sqrt{5} + 5\sqrt{5} + 3\sqrt{5}\).

Combine like terms

We can combine these like terms to reach the final simplified expression. So, \(4\sqrt{5} + 5\sqrt{5} + 3\sqrt{5} = 12\sqrt{5}\).

Key concepts

Radicals

In mathematics, radicals involve the use of the radical symbol \( \sqrt{} \) to represent the square root or higher-order root of a number. Square roots, which answer the question 'What number squared gives us our original number?', require understanding of perfect squares such as \( \sqrt{4} = 2 \) because \( 2^2 = 4 \).

Understanding radicals is more than just memorizing perfect squares, it’s about recognizing how to break down complex numbers like 125 into simpler factors inside the radical. For example, \( \sqrt{125} \) can be broken down into \( \sqrt{25\times5} \) because 25 is a perfect square that yields a whole number when square rooted. Simplifying radicals can often make other operations in mathematics, such as addition and multiplication, much easier.

Combine Like Terms

When you encounter expressions with multiple terms under the radical sign, such as \( 4\sqrt{5} + 5\sqrt{5} + 3\sqrt{5} \), you may notice that each term is a multiple of \( \sqrt{5} \). These terms are considered 'like terms' because they all contain the same radical component.

To combine them, you simply add their numerical coefficients, much like you would add variables in algebraic expressions. In our case, you add 4, 5, and 3 to get 12, rendering the simplified form as \( 12\sqrt{5} \). This process is critical to simplifying expressions by reducing clutter and making them more manageable to work with.

Factoring Square Roots

Factoring square roots are a key step in simplifying radicals. The goal is to find the highest square factor of a given number in the radicand (the number under the radical). For instance, in \( \sqrt{125} \) you would look for the largest perfect square that divides 125, which is 25.

Therefore, \( \sqrt{125} \) is factored into \( \sqrt{25} \times \sqrt{5} \) and then simplified to \( 5\sqrt{5} \) since \( \sqrt{25} = 5 \). This process not only works for perfect squares but also when dealing with larger radicands where the square root isn't immediately obvious. Factoring them down exposes the hidden perfect squares and brings us closer to simplification.

Simplify Expressions

The process of simplifying expressions involves reducing them to their simplest form while retaining their original value. This is achieved through a series of steps, such as factoring square roots, combining like terms, and performing arithmetic operations.

In the context of radical expressions, after factoring to expose any perfect squares and taking their roots, we simplify the expression by combining like terms. With our example, once \( \sqrt{125} \) and \( \sqrt{45} \) are factored and simplified to \( 5\sqrt{5} \) and \( 3\sqrt{5} \) respectively, the expression is then combined, resulting in \( 12\sqrt{5} \). Simplifying is a powerful tool that can turn an intimidating equation into something much more manageable and understandable.