Problem 68 - Algebra A Combined Function

Question

Assume that all variables represent positive real numbers. $$ \sqrt{9 x^{4} y^{6}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Expression

The given expression is \( \sqrt{9 x^{4} y^{6}} \). The objective is to simplify this square root expression by finding equivalent forms.

Step 2: Factor Inside the Square Root

Recognize that the number 9 is a perfect square and can be factored as \( 3^2 \). This means that the expression inside the square root is \( 9 x^4 y^6 = (3^2)(x^4)(y^6) \).

Step 3: Apply Square Root to Each Factor Separately

Using the property \( \sqrt{a \cdot b \cdot c} = \sqrt{a} \cdot \sqrt{b} \cdot \sqrt{c} \), we separate the expression to \( \sqrt{3^2} \cdot \sqrt{x^4} \cdot \sqrt{y^6} \).

Step 4: Simplify Each Square Root Component

Calculate each component: \( \sqrt{3^2} = 3 \), \( \sqrt{x^4} = x^2 \), and \( \sqrt{y^6} = y^3 \). This is because \( (x^2)^2 = x^4 \) and \( (y^3)^2 = y^6 \).

Step 5: Combine the Simplified Terms

Combine the terms from the previous step to get the simplified form: \( 3x^2y^3 \).

Key Concepts

Perfect Squares

In mathematics, a perfect square is an integer that can be expressed as the square of another integer. For example, numbers like 1, 4, 9, and 16 are perfect squares, as they result from squaring the integers 1, 2, 3, and 4, respectively. Recognizing perfect squares is essential when simplifying radical expressions, as it allows you to break down complex expressions into simpler forms.
In the given exercise, we have the number 9 which is a perfect square because it equals \(3^2\). Perfect squares are important because when taking the square root, you are essentially reversing the squaring process. Thus, \(\sqrt{9} = 3\).

• Spotting perfect squares can significantly simplify solving equations or expressions involving radicals.
• This simplification process reduces a radical expression into its simplest form.

Square Root Properties

The properties of square roots make working with radicals much more manageable. One essential property is that the square root of a product equals the product of the square roots. Mathematically, this is expressed as \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\). This property is invaluable when simplifying expressions under the radical sign.
In our exercise, the expression \(\sqrt{9 x^{4} y^{6}}\) is simplified by applying this property. You break down the expression into its components: \(\sqrt{9} \cdot \sqrt{x^{4}} \cdot \sqrt{y^{6}}\). This technique separates each part, allowing individual simplification of each term:

• \(\sqrt{9} = 3\) as discussed in perfect squares.
• \(\sqrt{x^{4}} = x^2\) because squaring \(x^2\) returns \(x^4\).
• \(\sqrt{y^{6}} = y^3\), as \(y^3\) squared gives \(y^{6}\).

These properties help in reducing complex radical expressions into simpler, more understandable forms.

Algebraic Expressions

An algebraic expression consists of variables, numbers, and operations. It is crucial to understand how to manipulate these expressions when simplifying them. In cases involving square roots, variables often appear with exponents, just like \(x^4\) and \(y^6\) in our example.
The process of simplification involves understanding how exponents work within radicals. A key step is recognizing when an expression under a square root can be turned into a perfect square of an algebraic expression. For instance:

• \(x^4\) is \((x^2)^2\), indicating it is a perfect square.
• Likewise, \(y^6\) is \((y^3)^2\).

This insight is useful for breaking down complex expressions into manageable parts. The combination of recognizing perfect squares, using square root properties, and effectively working with algebraic expressions simplifies the entire process, leading to the final result of \(3x^2y^3\).
By tackling each aspect methodically, you can confidently simplify similar radical expressions in algebra and beyond.