Problem 61 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. (Lesson 9.3) $$ 9 \sqrt{36} $$

Step-by-Step Solution

Answer

The simplified form of \(9\sqrt{36}\) is 54.

Step 1: Identify the square root

Firstly, determine the square root of the number 36. The square root of 36 is 6.

Step 2: Multiply the numbers

Now, multiply the square root of 36( which is 6) by 9 to get the answer. So, the product would be \(9\times 6 = 54\).

Step 3: Simplify and Conclude

The simplified form of \(9\sqrt{36}\) is 54.

Key Concepts

Square Root Operations

Understanding square root operations is pivotal in simplifying radical expressions. A square root, represented by the symbol \( \sqrt{\phantom{x}} \), asks for a number which, when multiplied by itself, will produce the given number under the root symbol. For example, the square root of \(36\) is \(6\), because \(6 \times 6 = 36\).

When dealing with perfect squares like \(36\), the operation is straightforward as they have an integer as their square root. However, it becomes tricker with non-perfect squares, which might result in an irrational number that can't be neatly expressed and might have to be approximated or left in radical form. Recognizing perfect squares and knowing their roots by heart is a helpful skill in algebra that speeds up the simplification process.

Multiplying Radicals

Multiplying radicals might initially seem daunting, but with a bit of practice, it becomes a quick and mechanical process. The key is to remember that the product of two square roots, \( \sqrt{a} \times \sqrt{b} \), is the square root of the product of the numbers, or \( \sqrt{ab} \). However, when we have a number outside the radical, such as \( 9 \sqrt{36} \), the process is even simpler: we first simplify the radical and then multiply it by the number outside.

In our example, we started by simplifying \( \sqrt{36} \) which is \(6\), and then we multiplied it by \(9\). This is like multiplying \(9\) by \(6\) to get \(54\). Recognizing the situation allows for faster evaluation without having to do extra steps such as foiling or looking for common factors inside the radicals.

Algebraic Simplification

Algebraic simplification encompasses various methods that make expressions easier to work with or understand. In the context of radical expressions, simplification often involves reducing the expression to its most basic form without a radical sign, if possible. This involves recognizing perfect squares, cubes, and so on within the radical that can be taken out.

In the exercise \( 9 \sqrt{36} \), the simplification was straightforward since \(36\) is a perfect square. This scenario represents the ideal situation in algebraic simplification with radicals: when the number under the radical is a perfect square and thus allows for the complete removal of the square root. Developing a quick eye for such simplifications, where numbers are reduced to their most elementary components, significantly eases advancing through algebra problems and avoiding errors that can occur during the manipulation of complex expressions.