Problem 70 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ 9 \sqrt{18}-2 \sqrt{8} $$

Step-by-Step Solution

Answer

The simplified expression is \( 23 \sqrt{2} \).

Step 1: Simplify Each Square Root Separately

Start by simplifying each square root term in the expression. \( \sqrt{18} \) can be broken down as \( \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3 \sqrt{2} \). Similarly, \( \sqrt{8} \) can be rewritten as \( \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \sqrt{2} \).

Step 2: Substitute Simplified Roots

Replace \( \sqrt{18} \) and \( \sqrt{8} \) with their simplified forms. The expression becomes \( 9 \times 3 \sqrt{2} - 2 \times 2 \sqrt{2} \).

Step 3: Multiply Coefficients by Roots

Calculate the products by multiplying the coefficients by their respective roots. Therefore, \( 9 \times 3 \sqrt{2} = 27 \sqrt{2} \) and \( 2 \times 2 \sqrt{2} = 4 \sqrt{2} \).

Step 4: Combine Like Terms

Combine the simplified terms: \( 27 \sqrt{2} - 4 \sqrt{2} \). Subtract the coefficients of \( \sqrt{2} \) to get \( (27 - 4) \sqrt{2} = 23 \sqrt{2} \).

Key Concepts

Square Root Simplification

Square root simplification involves breaking down a square root into a product of simpler square roots. In our example, we look at expressions like \( \sqrt{18} \). We aim to express them in simpler forms by finding perfect square factors. This is because perfect squares have whole numbers as their square roots, making the expression easier to handle.
To simplify \( \sqrt{18} \), notice that 18 can be broken down into \( 9 \times 2 \). Here, 9 is a perfect square, and \( \sqrt{9} \) equals 3. We can rewrite \( \sqrt{18} \) as \( \sqrt{9} \times \sqrt{2} \), simplifying it to \( 3 \sqrt{2} \).
Similarly, \( \sqrt{8} \) can be rewritten as \( \sqrt{4 \times 2} \). Since \( \sqrt{4} \) equals 2, this becomes \( 2 \sqrt{2} \).
By handling square roots this way, we simplify the algebra, making it easier to work with in larger expressions.

Like Terms in Algebra

Like terms in algebra are terms that have the exact same variable part. Simplifying an expression involves identifying and combining these terms to make calculations easier. For the expression \( 9 \sqrt{18} - 2 \sqrt{8} \), after simplifying the square roots, it becomes \( 27 \sqrt{2} - 4 \sqrt{2} \).

• Both parts of the expression now include \( \sqrt{2} \), making them like terms.
• This allows us to combine them by adding or subtracting their coefficients, just like we would with algebraic variables.

Combining like terms simplifies the expression to \( 23 \sqrt{2} \), illustrating the importance of grouping similar elements to streamline the process.

Coefficients in Algebraic Expressions

Coefficients are the numerical parts of terms in algebraic expressions. They multiply the variables or constants in the expression. In the expression \( 9 \times 3 \sqrt{2} - 2 \times 2 \sqrt{2} \), the coefficients are the numbers in front of \( \sqrt{2} \).

• In \( 27 \sqrt{2} \), 27 is the coefficient.
• In \( 4 \sqrt{2} \), 4 is the coefficient.

Understanding coefficients is crucial when combining terms. By knowing they represent a scaling factor, you can effectively manipulate, add, or subtract terms. This skill is critical in simplifying expressions, solving equations, and ensuring calculations are accurate. Adding the coefficients of like terms results in simplifications like \( (27 - 4) \sqrt{2} = 23 \sqrt{2} \), demonstrating the power of coefficients in algebra.