Question

Simplify the radical expression. $$9 \sqrt{36}$$

Short answer

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

Step-by-step solution

Simplify The Number Outside The Square Root

The number outside of the square root is 9, which is already in its simplest form.

Simplify The Expression Under The Square Root

The expression under the square root is 36, the square root of 36 is 6.

Combine The Results

The final step is to multiply the number outside of the square root (9) with the simplified expression under the square root (6). The result is 9 * 6 = 54.

Key concepts

Square Roots

Understanding square roots is foundational in simplifying radical expressions. A square root, represented by the radical symbol \( \sqrt{} \), asks the question: 'What number, when multiplied by itself, will give the original number under the radical?' For example, the square root of 36 is 6 because \( 6 \times 6 = 36 \).

When you encounter a square root, the aim is to find this number, which is known as the 'radical' or 'root'. Perfect squares, like 4, 9, 16, 25, and so on, make finding the square root straightforward because they have whole numbers as their roots. However, when the number isn't a perfect square, you'll need to use different strategies to simplify the expression, such as factoring out squares or estimating the root.

Being comfortable with square roots is key because they're not only a part of radical simplification but are also involved in quadratic equations, Pythagorean theorem applications, and understanding the concept of irrational numbers.

Radical Simplification

Radical simplification is the process of making a radical expression as simple as possible. This often involves removing any perfect square factors that are inside the radical. For instance, in the example \(9 \sqrt{36}\), the number 36 is under the square root sign. Since 36 is a perfect square, we can take its square root, resulting in the number 6.

To simplify the radical further, any coefficients outside the square root should be evaluated or simplified if necessary. In our original exercise, the number outside the square root is 9. The simplification process tells us that the 9 stands alone, and it should now be multiplied by the simplified radical, resulting in the product of 9 and 6 which equals 54.

An important factor in radical simplification is recognizing perfect square factors within larger numbers that are not perfect squares themselves. This often involves breaking down the number into its prime factors and pairing the primes that form a perfect square. The pairs move outside the radical sign as their square root, simplifying the expression.

Arithmetic Operations

Arithmetic operations, including addition, subtraction, multiplication, and division, are the building blocks of many mathematical concepts, including the manipulation of radical expressions. In the context of our exercise, once the radical is simplified, we implement basic multiplication.

The simplified expression \(6\) which was derived from the square root of 36, is then multiplied by the coefficient outside, which is \(9\). This multiplication is a straightforward arithmetic operation: \(9 \times 6 = 54\). The operation illustrates how multiplication is used to combine the results of radical simplification, yielding the final simplified expression.

In other scenarios, you might have to add or subtract square roots, which can only be done when the radicals are like terms (i.e., they have the same number under the radical sign). Division may also come into play when you have a fraction under a radical. Understanding how to apply these operations to radical expressions and simplify the results is a valuable skill in algebra.