Problem 60 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ 9 \sqrt{100} $$

Step-by-Step Solution

Answer

The simplest form of \(9 \sqrt{100}\) is 90.

Step 1: Identify the Radicand

The radicand is the value inside the square root, which is 100 in this case.

Step 2: Calculate the square root of the radicand

The square root of 100 is 10. The square root is the value that, when multiplied by itself, gives the original number. In this case, \(10 \times 10\) equals 100.

Step 3: Multiply by the Coefficient

With the square root of the radicand found, multiply this value by the coefficient outside the square root, which is 9. So, we compute \(9 \times 10 = 90\).

Key Concepts

Radicand

The term 'radicand' refers to the number or expression that is inside the radical sign. In the context of square roots, the radical sign looks like a check mark with a line running along the top, and the radicand is nestled underneath. For instance, in the expression \(\sqrt{100}\), 100 is the radicand.

Understanding the radicand is crucial when simplifying radical expressions, as the simplification process often involves factoring the radicand into its prime factors and identifying perfect squares. A perfect square is a number that has an integer as its square root, like 100, which is the square of 10. When simplifying, any square factors are taken out from under the radical sign, leaving a simplified radicand or none at all if the entire radicand is a perfect square.

Square Root

The square root is a mathematical operation that answers the question: 'Which number, when multiplied by itself, will give the original number?' It's represented by the radical sign, \(\sqrt{}\). For your exercise, the square root of 100 is sought.

Calculating the square root is essential when working with radical expressions. In some cases, like with 100, the square root is straightforward, as 100 is a perfect square. However, the square roots of non-perfect squares would result in an irrational number, which may need to be approximated or left in radical form. In simplifying radical expressions, finding the square root of the radicand when possible alleviates the complexity of the expression.

Coefficient

In algebra, a coefficient is a number that multiplies a variable or, as in this instance, a square root. Think of it as the 'scaling factor' that determines how many times to count the associated value. In the provided example, \(9\sqrt{100}\), the number 9 is the coefficient.

Identifying the coefficient is key because it affects the value of the entire expression once the square root is calculated. After finding the square root of the radicand, you multiply it by the coefficient to get the final simplified form. If the coefficient is 1, it doesn't change the value and is usually not written. In some cases, if there's a negative coefficient, it indicates that the expression is the opposite of the positive counterpart.

Arithmetic Operations

Arithmetic operations are the fundamental processes of addition, subtraction, multiplication, and division. In simplifying radical expressions, multiplication and division are typically used.

Once the square root has been determined, as with the radicand '100' becoming '10', that result is then multiplied by the coefficient outside the radical, in this case, 9, using arithmetic multiplication. This scenario exemplifies how simplifying square roots often involves a two-step process: first, square root calculation, and then arithmetic operation with the coefficient. Mastery of these operations ensures accurate simplification of radical expressions and is an essential skill in algebra.