Problem 34 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ 8 \sqrt{25 y^{3}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression: \(8 \sqrt{25 y^{3}}\) Answer: \(40y \sqrt{y}\)

Step 1: Identify the expression

Our given expression is: $$ 8 \sqrt{25 y^{3}} $$

Step 2: Simplify the number inside the square root

We have \(25\) inside the square root, which can be simplified to \(5\) as \(5^2 = 25\). $$ 8 \sqrt{25 y^{3}} = 8 \cdot 5 \sqrt{y^{3}} $$

Step 3: Simplify the variable inside the square root

We now have \(y^{3}\) inside the square root. Since the square root of \(y^{2}\) is \(y\), we can simplify the expression further by taking the square root of \(y^{2}\) (which is part of \(y^3\)) outside of the square root and leaving the remaining \(y\) inside. $$ 8 \cdot 5 \sqrt{y^{3}} = 8 \cdot 5 \cdot y \sqrt{y} $$

Step 4: Multiply the constants to get the final answer

Multiply the constants outside the square root: $$ 8 \cdot 5 \cdot y \sqrt{y} = 40y \sqrt{y} $$ The simplified radical expression is: \(40y \sqrt{y}\).

Key Concepts

Square Root

Understanding the square root is fundamental in simplifying radical expressions. Essentially, a square root answers the question: what number, when multiplied by itself, equals the given number? For instance, the square root of 9, represented as \( \sqrt{9} \), is 3, because \( 3 \times 3 = 9 \).

When simplifying square roots, we look for perfect squares, which are numbers like 1, 4, 9, 16, 25, and so on, as these have whole numbers as their square roots. In the exercise \( 8 \sqrt{25 y^{3}} \), we see that 25 is a perfect square. That's why we can replace \( \sqrt{25} \) with 5. For non-perfect squares, the process requires finding the prime factors and pairing them to simplify the expression.

Radical Simplification

Radical simplification involves breaking down the expression under the radical sign to its simplest form. To do this, one must identify and separate the perfect squares (or cubes for cube roots, and so on) from the rest of the terms. For instance, in the expression \( \sqrt{y^3} \), \( y^3 \) can be factored into \( y^2 \) and \( y \). Since we know \( \sqrt{y^2} = y \), we can simplify the expression to \( y \sqrt{y} \).

This process reduces the complexity of the expression and makes it easier to handle in equations and applications. The main goal is to always have the expression under the radical sign be free of perfect powers to avoid redundancy and to keep it as reduced as possible.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Simplifying such expressions is a fundamental skill in algebra, allowing us to solve equations and inequalities more efficiently. When working with radicals within algebraic expressions, the properties of radicals and arithmetic operations come into play.

In the provided exercise, the algebraic expression involving a radical is simplified by dealing with each part separately: numbers and variables. With numbers, we simplify the coefficient and the constant inside the square root as shown in the solution. With variables, we look for exponents that are multiples of the index of the radical (which is 2 for square roots) to simplify them accordingly. In our example, we simplified \( y^3 \) within the square root by recognizing that \( y^2 \) is a perfect square and could be separated from the radical. The ulterior motive behind simplifying expressions, including those with radicals, is to make the equations or expressions more manageable and to clearly identify solutions.