Problem 117 - Elementary and Intermediate Algebra

Question

Simplify each expression. Assume that all variables represent positive numbers. $$4 \sqrt{48 y^{3}}-3 y \sqrt{12 y}$$

Step-by-Step Solution

Answer

The simplified expression is \(10y \sqrt{3y}\).

Step 1: Simplify the square roots

First, let's simplify the square roots in the expression. For \(\sqrt{48y^3}\):- Break it down: \(48y^3 = 16 \times 3 \times y^2 \times y = 4^2 \times 3 \times y^2 \times y\).- Use the property \(\sqrt{a^2} = a\) for simplification: \(\sqrt{16} = 4\) and \(\sqrt{y^2} = y\).- So, \(\sqrt{48y^3} = 4 \times y \times \sqrt{3y} = 4y\sqrt{3y}\).For \(\sqrt{12y}\):- Break it down: \(12y = 4 \times 3 \times y = 2^2 \times 3 \times y\).- So, \(\sqrt{12y} = 2\sqrt{3y}\).

Step 2: Substitute simplified square roots into the expression

Substitute the simplified square roots back into the expression:\[4 \sqrt{48y^3} - 3y \sqrt{12y} = 4 \times 4y \sqrt{3y} - 3y \times 2\sqrt{3y}\]

Step 3: Simplify the coefficients

Simplify the expression by multiplying the coefficients:\[4 \times 4y \sqrt{3y} = 16y \sqrt{3y}\]\[3y \times 2 \sqrt{3y} = 6y \sqrt{3y}\]

Step 4: Combine like terms

Now, combine the like terms, which both have \(y \sqrt{3y}\):\[16y \sqrt{3y} - 6y \sqrt{3y} = (16y - 6y) \sqrt{3y}\]\[= 10y \sqrt{3y}\]

Step 5: Final Answer

The simplified expression after combining like terms is: \[10y \sqrt{3y}\].

Key Concepts

Square Roots

Square roots are a fundamental concept in algebra that represent a value which, when multiplied by itself, gives the original number. Understanding square roots is essential for the simplification of expressions.
When simplifying square roots involving numbers and variables, it helps to break them down into smaller factors. For instance, in the expression \(\sqrt{48y^3}\), we decompose it as \(48y^3 = 16 \times 3 \times y^2 \times y\), which simplifies to \(4^2 \times 3 \times y^2 \times y\). This allows us to use the property that \(\sqrt{a^2} = a\), simplifying \(\sqrt{16} = 4\) and \(\sqrt{y^2} = y\). Thus, \(\sqrt{48y^3}\) simplifies to \(4y\sqrt{3y}\).
Similarly, for \(\sqrt{12y}\), we see \(12y = 4 \times 3 \times y = 2^2 \times 3 \times y\), simplifying it to \(2\sqrt{3y}\). Understanding how to break down and simplify square roots allows complex expressions to be managed more easily.

Like Terms

In algebra, like terms refer to terms that contain the same variables raised to the same powers. This principle is crucial when simplifying expressions, as it allows combined arithmetic operations only on like terms.
For example, the expression we dealt with included terms like \(16y\sqrt{3y}\) and \(6y\sqrt{3y}\). These are like terms because they both consist of the variable \(y\) and the same square root \(\sqrt{3y}\). By identifying like terms, we can combine them, thus simplifying the expression.
Combining like terms involves arithmetic on their coefficients. In our exercise, combining \(16y\sqrt{3y}\) and \(-6y\sqrt{3y}\) results in \((16y - 6y)\sqrt{3y}\), simplifying to \(10y\sqrt{3y}\). This step is pivotal in reaching the final, simplified form of an expression.

Coefficients

Coefficients are the numerical parts of terms in an expression. They multiply the variables they accompany and are key to the simplification and manipulation of algebraic expressions.
In our exercise, the coefficients involved were manipulated directly to simplify the expression: \(4\) and \(3y\) in \(4\sqrt{48y^3} - 3y\sqrt{12y}\). When simplified, these coefficients transform to \(16y\) and \(6y\) respectively after incorporating their corresponding square roots, yielding \(16y\sqrt{3y}\) and \(6y\sqrt{3y}\).
Understanding coefficients is important because they determine how terms can be combined. They act as multipliers and play a crucial role in the overall arithmetic within the expression.

Positive Numbers Assumption

The assumption that all variables represent positive numbers helps simplify algebraic expressions, as it removes the need to consider multiple cases for different signs.
If variables could be negative, additional considerations for cases may be necessary, especially when dealing with square roots, as the negative values could change the behavior of the roots. However, with positive numbers, such issues do not arise.
Additionally, assuming positive numbers ensures valid real number solutions, particularly in simplifying expressions with square roots. This assumption underlines the steps taken in the exercise, ensuring that the result is always a real number without the complications that arise from negative values.