Problem 24 - Elementary and Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ 3 \sqrt{8 x}(2 \sqrt{2 x^{3} y}) $$

Step-by-Step Solution

Answer

The simplified expression is \(24x^2 \sqrt{y}\).

Step 1: Distribute the Multiplication

Distribute the multiplication over the expression: \ \( 3 \sqrt{8x} \times 2 \sqrt{2x^3 y} \). \ This becomes \ \( (3 \times 2) \times (\sqrt{8x} \times \sqrt{2x^3 y}) = 6 \times (\sqrt{8x} \times \sqrt{2x^3 y}) \).

Step 2: Multiply the Radicands

When multiplying square roots, you can multiply the numbers inside the radicands: \ \( \sqrt{8x} \times \sqrt{2x^3 y} = \sqrt{(8x) \times (2x^3 y)} = \sqrt{16x^4 y} \).

Step 3: Simplify the Square Root

Separate the radicand \( 16x^4y \) into perfect squares and simplify: \ \( \sqrt{16x^4 y} = \sqrt{16} \times \sqrt{x^4} \times \sqrt{y} \). \ Since \( \sqrt{16} = 4 \) and \( \sqrt{x^4} = x^2 \), it becomes \( 4x^2\sqrt{y} \).

Step 4: Combine the Expressions

Multiply the simplified radical expression by the coefficient: \ \( 6 \times 4x^2 \sqrt{y} = 24x^2 \sqrt{y} \). \ This gives the final simplified expression.

Key Concepts

Multiplication of Radicals

When you're multiplying radicals, such as square roots, you follow the rule of multiplying the numbers inside the radicals, also known as the radicands. If you have two square roots like \( \sqrt{a} \times \sqrt{b} \), you can combine them into a single square root: \( \sqrt{a \times b} \).
This is because the square root operation is distributive over multiplication. For example, with \( 3 \sqrt{8x} \times 2 \sqrt{2x^3 y} \), we can first focus on the numbers inside the radicals. Break it down like this:

• Multiply the coefficients: \( 3 \times 2 = 6 \).
• Multiply the radicands: \( \sqrt{8x} \times \sqrt{2x^3 y} = \sqrt{(8x) \times (2x^3 y)} = \sqrt{16x^4 y} \).

By multiplying these components correctly, you prepare for the simplification process, merging them into a single radical expression.
Practicing this approach strengthens your understanding of algebraic manipulations and radical multiplication.

Simplifying Radicals

Simplifying radicals involves reducing a radical expression to its simplest form. The idea is to remove perfect square factors from under the square root, turning them into whole numbers outside the radical.
Once we have a large radicand, like \( \sqrt{16x^4 y} \), our goal is to break it down into simpler parts:

• Identify perfect squares: In \( 16x^4 \), both \( 16 \) and \( x^4 \) are perfect squares.
• Extract these perfect squares: \( \sqrt{16} \) simplifies to \( 4 \) and \( \sqrt{x^4} \) simplifies to \( x^2 \).

This leaves the expression \( 4x^2 \sqrt{y} \), showcasing how perfect squares simplify algebraic expressions efficiently.
Regular practice can make simplification intuitive, helping you quickly spot perfect squares and streamline expressions.

Perfect Squares

Perfect squares are numbers or expressions raised to the second power. Recognizing them is crucial for simplifying radicals, as they help convert complicated expressions into manageable terms.
Consider \( 16 \) and \( x^4 \) from our example. Here's a deeper look:

• \( 16 \) is a perfect square because it equals \( 4^2 \).
• \( x^4 \) is \( (x^2)^2 \), hence a perfect square.

These are special cases where square rooting yields whole values, allowing for dramatic simplification.
By focusing on breaking down expressions into perfect squares, you can drastically simplify your work with radicals, increasing both accuracy and speed in solving algebraic problems.
Understanding perfect squares makes you adept at manipulating radicals, an essential skill in algebra.