Problem 48 - Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \frac{\sqrt{75 y^{3}}}{\sqrt{3 y}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 5y \).

Step 1: Simplify the Square Roots

Start by simplifying the square roots in the numerator and the denominator separately. For the numerator \( \sqrt{75 y^3} \):- Break down the numbers and variables under the square root: \( 75 = 25 \times 3 \), so \( \sqrt{75 y^3} = \sqrt{25} \times \sqrt{3} \times \sqrt{y^3} \). - Simplify: \( \sqrt{25} = 5 \) and \( \sqrt{y^3} = y^{1.5} = y^{1} \times \sqrt{y} \). - Thus, \( \sqrt{75 y^3} = 5y\sqrt{3y} \).For the denominator \( \sqrt{3y} \): it remains as is because it's already simplified.

Step 2: Cancel Common Terms

Now that you have \( \frac{5y\sqrt{3y}}{\sqrt{3y}} \), we need to simplify further by canceling out common terms in the square root:- Recognize that \( \sqrt{3y} \) is common to both the numerator and the denominator.- Cancel \( \sqrt{3y} \) in both the numerator and the denominator.This leaves us with \( 5y \).

Key Concepts

Simplifying Square Roots

Simplifying square roots allows us to make expressions easier to handle. When faced with the task of simplifying square roots, focus on breaking down each component into parts that are easy to work with.

In the expression \(\frac{\sqrt{75 y^{3}}}{\sqrt{3 y}}\), we need to simplify \(\sqrt{75 y^3}\) in the numerator first. Breaking it down involves factoring the number 75 into perfect squares that are easier to simplify. Notice that 75 is divisible by 25, which itself is a perfect square. Therefore, \(\sqrt{75}\) can be rewritten as \(\sqrt{25 \times 3}\).

Now it becomes clear: \(\sqrt{25} = 5\), a neat whole number. Similarly, when simplifying \(\sqrt{y^3}\):

• We interpret it as \(\sqrt{y^2 \times y}\). The square root of \(y^2\) is \(y\).
• Putting it together, \(\sqrt{y^3} = y \times \sqrt{y}\).

Combining these with the simplified number section gives us \(\sqrt{75 y^3} = 5y\sqrt{3y}\). Highlighting common factors in the numerator and denominator helps in further simplification.

Positive Real Numbers

Positive real numbers refer to numbers greater than zero, including all positive fractions and decimals. In mathematics, when we talk about handling variables, the statements often assume they represent positive real numbers.

In this exercise, assuming all variables are positive ensures that we don't encounter undefined expressions. For instance, when simplifying a square root like \(\sqrt{y^3}\), knowing that \(y\) is positive helps us avoid complex numbers that arise with negative values in real-world scenarios.

Handling positive values simplifies the rules and applications, ensuring that each step remains manageable and logical. Variables under square roots, for instance, must be non-negative, as taking the square root of a negative number leads to imaginary results, stretching beyond basic real number operations.

Understanding these properties of positive real numbers gives clarity and confidence when dealing with expressions that include multiple radical terms.

Mathematical Notation

Mathematical notation is a way to represent numbers and operations using symbols and formulas. It provides a universal language to communicate complex mathematical concepts clearly and concisely.

This exercise uses several symbols: the square root symbol \(\sqrt{}\), variables \(y\), and the fraction bar, which denotes division. Each carries its challenges and significance.

For example, the square root sign \(\sqrt{}\) denotes finding a number which, when multiplied by itself, gives the original number inside the root. The division sign represented by the fraction bar in \(\frac{\sqrt{75 y^{3}}}{\sqrt{3 y}}\) shows a division of the numerator by the denominator, emphasizing the importance of simplifying expressions through operations like canceling common terms.

Maintaining consistency and accuracy in notation involves careful placement and understanding the role of each symbol. This leads to clear, effective communication and eases the simplification process across various mathematical expressions.