Problem 21 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{32 x^{2} y^{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \( 4xy\sqrt{2y} \).

Step 1: Prime factorize the number

First, we need to factor the number under the square root: \( 32 \). This can be expressed as \( 32 = 2^5 \).

Step 2: Write the expression inside the square root

Write the expression inside the square root using the prime factorization and variables: \( \sqrt{2^5 \cdot x^2 \cdot y^3} \).

Step 3: Simplify using properties of square roots

Apply the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). Separate the factors: \( \sqrt{2^5} \cdot \sqrt{x^2} \cdot \sqrt{y^3} \).

Step 4: Simplify each radical

Simplify each radical: \( \sqrt{2^5} = 2^2 \sqrt{2} = 4\sqrt{2} \), \( \sqrt{x^2} = x \), and \( \sqrt{y^3} = y\sqrt{y} \).

Step 5: Combine the simplified terms

Now, combine all the simplified terms: \( 4x y \sqrt{2y} \). This is the simplified expression where no further factor can be taken out of the square root.

Key Concepts

Prime Factorization

Prime factorization is a crucial step when dealing with radicals. It involves breaking down a number into its basic building blocks, which are prime numbers. A prime number is a natural number greater than 1 that is not divisible by any other numbers except for 1 and itself. For the number 32, the process goes like this:

• 32 is even, so divide by 2 to get 16.
• 16 is also even, divide by 2 to get 8.
• 8 continues to be divisible by 2, giving 4.
• 4 divides by 2 to get 2, and finally, 2 is already prime.

This means 32 = 2 × 2 × 2 × 2 × 2, or using exponential notation, 32 = 2^5. Breaking numbers into their prime components makes it easier to simplify expressions, especially when under a square root.

Properties of Square Roots

Understanding and using the properties of square roots are vital for simplifying radical expressions. One such property states that the square root of a product equals the product of the square roots of the factors: \[\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\]This property allows you to handle complex roots more efficiently. Suppose you have \(\sqrt{32x^2y^3}\). To make things simpler, you break this into parts:

• \( \sqrt{32} \) using our prime factorization becomes \( \sqrt{2^5} \)
• \( \sqrt{x^2} \) and \( \sqrt{y^3} \) represent the variable components.

Applying the property helps separate and simplify each segment individually.

Expression Simplification

Expression simplification combines several techniques to make a mathematical expression more manageable. Once numbers and variables are factored and roots are taken, simplification is the next logical step. Consider \(\sqrt{32x^2y^3}\). After applying knife-edge prime factorization and the properties of square roots, we rewrite this as:

• \( \sqrt{2^5} \) is equivalent to \(2^2 \sqrt{2} = 4\sqrt{2}\).
• \( \sqrt{x^2} \) simplifies directly to \(x\).
• \( \sqrt{y^3} \) factors into \(y\sqrt{y}\).

The simplified expression \(4xy\sqrt{2y}\) embodies these principles. It's now clear at a glance, respecting order and making further calculations straightforward.

Mathematical Operations

At the heart of simplifying expressions lie mathematical operations. These operations interconnect processes we use in breaking down, rearranging, and simplifying radical expressions. They are the bridge between complex expressions and simplified results.In our original problem, fundamental operations include multiplication and the distributive property seen when applying roots:

• Multiplication - used to combine factors under a single radical (\(32x^2y^3\)).
• Distributive property of square roots - allowing \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) to separate and simplify parts.

Each operation is a puzzle piece that makes up the larger picture of the simplified expression \(4xy\sqrt{2y}\). Mastery of these operations is necessary for confidently handling more complex equations in the future.