Problem 86 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ -\sqrt{4} \cdot \frac{\sqrt{81}}{\sqrt{36}} $$

Step-by-Step Solution

Answer

-3

Step 1: Simplify Each Square Root

Simplify each term in the given equation separately. The square root of 4 is 2, the square root of 81 is 9 and the square root of 36 is 6, as they are all perfect squares. The equation will therefore transform into: -2 * (9/6)

Step 2: Perform Multiplication and Division

Now perform the multiplication and division of the simplified result to obtain the answer. Multiplication should be performed before division, following the BODMAS rule (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This results in: -(2 * 9)/6

Step 3: Simplify the Expression

The result from step two is -18/6. Simplify the division to find the simplest form of the given expression.

Key Concepts

Simplify Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, of numbers. Simplifying a radical expression means rewriting it in the simplest form possible.

• Identify and simplify radical terms. For example, \(\sqrt{4} = 2\), because 4 is a perfect square.
• The goal is to reduce the expression so that there are no more roots in the denominator or as coefficients where unnecessary.

This simplification helps in performing mathematical operations easily. For instance, the expression \(-\sqrt{4} \cdot \frac{\sqrt{81}}{\sqrt{36}}\) is simplified by dealing with each radical individually.\(\sqrt{4}\) simplifies to 2, \(\sqrt{81}\) simplifies to 9, and \(\sqrt{36}\) simplifies to 6 because these numbers are perfect squares. As a result, the expression transforms to \(-2 \cdot \frac{9}{6}\). This form is far easier to handle for further arithmetic operations.

Perfect Squares

Perfect squares are numbers that have whole numbers as their square roots. Understanding which numbers are perfect squares helps tremendously in simplifying radical expressions.

• A perfect square is basically \(n^2\). For instance, 4, 9, and 36 are perfect squares since their square roots are whole numbers: 2, 3, and 6, respectively.
• Recognizing perfect squares allows you to take roots effortlessly.

To take advantage of perfect squares in expressions, look for terms that can be simplified to whole numbers. This not only reduces complexity but makes arithmetic operations like addition, subtraction, multiplication, and division much more straightforward. By identifying \(\sqrt{4}, \sqrt{81},\) and \(\sqrt{36}\) as 2, 9, and 6, respectively, one bypasses unnecessary calculations and clears the expression for further processing.

BODMAS Rule

The BODMAS rule is an order of operations guideline ensuring calculations are carried out correctly. It stands for Brackets, Orders (like roots and powers), Division and Multiplication, and Addition and Subtraction.

• Perform operations inside brackets first. In our example, the expression is viewed as a series of operations to simplify.
• Next, simplify radicals as they are 'Orders' by this rule.
• Proceed with Division and Multiplication from left to right. In our case: \(-2 \cdot \frac{9}{6}\), calculate \(-18/6\).

Understanding and applying the BODMAS rule prevents errors and makes expressions easier to tackle. In this scenario, following the BODMAS order: first addressing roots, then moving onto the division and multiplication steps, gives us the final simplest form of the expression. Adhering to BODMAS ensures clarity and accuracy in mathematical procedures.