Problem 62 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify perfect squares

To simplify the radical \( \sqrt{9 x^{4} y^{6}} \), start by identifying the perfect squares inside the radical. Notice that 9, \( x^4 \), and \( y^6 \) are all perfect squares. We have \( \sqrt{9} = 3 \), \( \sqrt{x^4} = x^2 \), and \( \sqrt{y^6} = y^3 \).

Step 2: Apply the square root to each perfect square

Apply the square root to each perfect square separately. This means the expression becomes \( \sqrt{9} \times \sqrt{x^4} \times \sqrt{y^6} \). This simplifies to \( 3 \times x^2 \times y^3 \).

Step 3: Write the simplified radical expression

Combine the simplified terms from the previous step to write the simplified expression as one term. Thus, the simplified form of \( \sqrt{9 x^4 y^6} \) is \( 3x^2y^3 \).

Key Concepts

Perfect Squares

Perfect squares are numbers or expressions that are the square of an integer or another expression. Essentially, if you multiply a number by itself and it results in the original number, you have a perfect square. For example, 9 is a perfect square because it equals the square of 3, since

• \(3 \times 3 = 9\)
• \(9 = 3^2\)

Similarly, in algebraic terms, \(x^4\) is a perfect square because it can be written as

• \((x^2)^2\)

you'll notice that if you take the square of \(x^2\), you end up with \(x^4\). Perfect squares are critical in simplifying radicals because they allow us to "take out" the square root of terms easily.

• Number: 4, 16, and 25 are all perfect squares
• Algebraic: \(y^6\) since \((y^3)^2 = y^6\)

Recognizing perfect squares helps simplify calculations and expressions efficiently.

Square Root

The square root of a number or expression is a value that, when multiplied by itself, gives the original number. This operation is denoted by the radical symbol \(\sqrt{}\). For example, the square root of 9 is 3 because

• \(3 \times 3 = 9\)

In algebra, knowing how to compute the square root of expressions allows us to simplify complex radicals. If an expression under the radical sign includes perfect squares, you can simplify it efficiently.

• The square root of \(x^4\) is \(x^2\) because \( (x^2)^2 = x^4\).
• The square root of \(y^6\) is \(y^3\) since \( (y^3)^2 = y^6\).

Breaking down the original expression into its components helps in reversing the squaring process and simplifies the expression to a more manageable form.

Radical Expressions

Radical expressions involve the use of the radical symbol \(\sqrt{}\) to denote roots, the most common being square roots. Simplifying these expressions involves finding and extracting perfect square factors to simplify the entire radical as much as possible. Consider the expression

• \(\sqrt{9x^4y^6}\)

Breaking down this expression involves recognizing:

• \(\sqrt{9} = 3\)
• \(\sqrt{x^4} = x^2\)
• \(\sqrt{y^6} = y^3\)

The factors \(9\), \(x^4\), and \(y^6\) are all perfect squares, making it easier to directly apply the square root to each factor. Simplifying radical expressions aims to reduce radicals to their simplest form. The objective is also to ensure no perfect square factor remains under the radical, streamlining the expression into its simplest terms, like transforming \(\sqrt{9x^4y^6}\) into \(3x^2y^3\).