Problem 31 - Algebra 2

Question

Simplify. $$ \sqrt{25 x^{4} y^{6}} $$

Step-by-Step Solution

Answer

The simplified form is \(5x^2y^3\).

Step 1: Identify Perfect Squares

First, we recognize which components within the radicand are perfect squares. In this expression, we have the number 25, the variable term \(x^4\), and the variable term \(y^6\). Each of these can be written as a square of a number or an expression. Namely, \(25 = 5^2\), \(x^4 = (x^2)^2\), and \(y^6 = (y^3)^2\).

Step 2: Apply the Square Root

The square root can now be applied separately to each perfect square component: \(\sqrt{25} = 5\), \(\sqrt{x^4} = x^2\), and \(\sqrt{y^6} = y^3\).

Step 3: Combine the Results

Finally, combine all the simplified components obtained from the square roots. This gives us: \(5x^2y^3\).

Key Concepts

Perfect Squares

Perfect squares are numbers or expressions that can be expressed as the square of another number or expression. Understanding perfect squares is crucial when simplifying expressions that involve square roots, as they allow us to directly compute their square roots easily.

• For example, the number 25 is a perfect square because it can be written as \(5^2\).
• Similarly, in algebraic terms, \(x^4\) is a perfect square because it can be rewritten as \((x^2)^2\), and \(y^6\) is \((y^3)^2\).

Recognizing perfect squares allows us to take advantage of the properties of square roots, making the simplification process straightforward.

Radical Expressions

A radical expression involves the use of the radical sign \(\sqrt{}\), which represents the root of a number or expression. Simplifying radical expressions is a common task in algebra, and it often involves identifying and simplifying perfect squares within the radicand (the number or expression inside the square root).

• To simplify, look out for numbers and expressions in the radicand that are perfect squares.
• Factor the radicand completely to find terms that can be "pulled out" of the square root.
• For example, in \(\sqrt{25 x^{4} y^{6}}\), each of the factors 25, \(x^4\), and \(y^6\) are perfect squares and simplify to \(5\), \(x^2\), and \(y^3\) respectively, when their square roots are taken.

Simplifying radical expressions not only makes them easier to work with but also helps in further algebraic operations.

Square Root Properties

The properties of square roots are essential tools for simplifying radical expressions. Understanding these properties makes the process of simplification logical and manageable.

• Product Property: \(\sqrt{a \, b} = \sqrt{a} \, \sqrt{b}\). This property allows us to split the square root of a product into the product of square roots, aiding in simplification.
• Simplification: When applying the product property, ensure each term inside the radical is a perfect square, if possible, so they can be simplified directly.
• Example in use: For \(\sqrt{25 x^{4} y^{6}}\), using the product property, we can separate it as \(\sqrt{25} \, \sqrt{x^4} \, \sqrt{y^6}\), and since each is a perfect square, they simplify directly to \(5\), \(x^2\), and \(y^3\) respectively.

Mastering square root properties empowers students to transform complex radical expressions into simpler ones, assisting their understanding and application in wider mathematical contexts.