Problem 71 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(2 \sqrt{4 y}-2 \sqrt{9 y}\)

Step-by-Step Solution

Answer

The simplified version of the given expression is \(-2y\)

Step 1: Simplify the square roots

The first step is to simplify each of the square roots in the expression. The square root of 4 is 2, and the square root of 9 is 3. Hence, the expression can be written as: \(2 *(2*y) - 2 *(3*y)\)

Step 2: Multiply terms

Next, simplify the equation by multiplying each term. The expression becomes: \(4y - 6y\)

Step 3: Combine like terms

The final step is to combine like terms. This simplifies the expression to: \(-2y\)

Key Concepts

Understanding Square Roots

Square roots are a fundamental concept in algebra and represent a number which, when multiplied by itself, gives the original number. They are denoted by the symbol \( \sqrt{} \) and can greatly simplify expressions.

For instance, in our example \( \sqrt{4y} \) and \( \sqrt{9y} \) are the square roots of \(4y \) and \(9y\) respectively. Recognizing that the square roots of perfect squares such as 4 and 9 are 2 and 3 helps in simplifying expressions quickly. The understanding that \( \sqrt{4} = 2 \) and \( \sqrt{9} = 3 \) is critical, as this makes further calculations straightforward.

Using this knowledge, the expression from our exercise simplifies the square roots to the form \(2y \) and \(3y\), by multiplying the extracted square roots with the accompanying variable, \(y\). Hence, square roots not only help in simplifying expressions but can also provide insight into the nature of numbers and variables within an expression.

Combining Like Terms

Combining like terms is a process used to simplify algebraic expressions. Like terms are terms that have the same variable raised to the same power. They can be combined by adding or subtracting their coefficients.

In the given exercise, after simplifying the square roots, we get terms \(4y\) and \(6y\). These terms are considered 'like terms' because they both contain the variable \(y\), raised to the same power, which is the first power or simply \(y\).

To combine them, you simply perform the arithmetic operation indicated: \(4y\) minus \(6y\) gives us \( -2y \). It's essential to maintain the signs in front of the coefficients, as they are part of the term's value. Mistakes in sign management can lead to incorrect results. Remember, combining like terms is a key step in simplifying algebraic expressions and obtaining the most reduced form of the expression.

Multiplying Algebraic Terms

Multiplying algebraic terms involves multiplying numbers, variables, or both, following certain rules. One of the most important rules to remember is the distributive property, which allows us to multiply a term outside a bracket to each term inside.

In our exercise, multiplying the coefficients outside the square roots with the simplified square roots themselves exemplifies this principle: \(2 \times 2y\) and \(2 \times 3y\). We focus on multiplying the numerical coefficients first—2 by 2, which gives 4, and 2 by 3, which yields 6.

The variable \(y\), which is the same in both terms, does not change during this step. It's important to ensure the variable is correctly carried through these operations. Consistency in keeping the variable part of the term undisturbed is key to accurate multiplication of algebraic terms. This step is crucial because it sets up the expression for the last simplification through combining like terms.