Problem 53 - Intermediate Algebra

Question

Simplify. See Examples 3 and 4 $$ -\sqrt{32 a^{8} b^{7}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-4a^4b^3\sqrt{2b}\).

Step 1: Prime Factorization

First, we will find the prime factorization of the number 32. 32 can be broken down into its prime factors as follows: \[ 32 = 2^5 \]

Step 2: Simplify the Radicand

The expression inside the square root is \(32a^8b^7\). Substitute the prime factorization from Step 1 into the expression:\[ 32a^8b^7 = (2^5)(a^8)(b^7) \]

Step 3: Apply the Square Root Property

Use the property that \( \sqrt{xy} = \sqrt{x} \cdot \sqrt{y} \) to simplify:\[ \sqrt{(2^5)(a^8)(b^7)} = \sqrt{2^5} \cdot \sqrt{a^8} \cdot \sqrt{b^7} \]

Step 4: Simplify Each Component

Now, simplify each square root individually:- \( \sqrt{2^5} = \sqrt{2^4 \times 2} = \sqrt{2^4} \cdot \sqrt{2} = 2^2 \cdot \sqrt{2} = 4\sqrt{2} \)- \( \sqrt{a^8} = a^4 \)- \( \sqrt{b^7} = \sqrt{b^6 \times b} = \sqrt{b^6} \cdot \sqrt{b} = b^3 \cdot \sqrt{b} = b^3\sqrt{b} \)

Step 5: Combine Simplified Components

Now combine the results from Step 4:\[ \sqrt{(2^5)(a^8)(b^7)} = 4\sqrt{2} \cdot a^4 \cdot b^3\sqrt{b} \]Multiply the terms together:\[ = 4a^4b^3 \sqrt{2b} \]

Step 6: Apply the Negative Sign

Multiply the entire expression by the negative sign from the original problem:\[ -\sqrt{32a^8b^7} = -4a^4b^3\sqrt{2b} \]

Key Concepts

Prime Factorization

When simplifying radicals, the first step often involves prime factorization. Prime factorization means breaking down a composite number into the product of its prime factors.

• For example, the number 32 can be expressed as a product of prime numbers: 32 equals \(2^5\), since 2 multiplied by itself five times equals 32.
• Prime factorization is useful because it reveals the simplest building blocks of a number, which is crucial for simplifying radicals.

Utilizing prime factors, especially when dealing with variables like in \(32a^8b^7\), helps you manage the components inside a radical systematically. Each element of the product (numbers and variables) can then be broken down similarly to more easily apply further mathematical operations.

Square Root Property

The square root property is a critical element when simplifying expressions under a radical sign, which helps break down complex expressions into smaller, more manageable pieces.

• According to the square root property, \( \sqrt{xy} = \sqrt{x} \cdot \sqrt{y} \). This property allows you to simplify the expression inside a square root by separating it into smaller parts that can be simplified individually.
• For instance, if you have \(\sqrt{32a^8b^7}\), you can apply this property to separate this into \(\sqrt{2^5} \cdot \sqrt{a^8} \cdot \sqrt{b^7}\).

This step makes it easier to simplify each square root term, often turning complex algebraic expressions into simpler formats.

Algebraic Expressions

Algebraic expressions comprise both numbers and variables linked by operations such as addition, subtraction, multiplication, and division.

• They are fundamental in expressing mathematical relationships and performing calculations.
• When handling radicals like in \(-\sqrt{32a^8b^7}\), understanding the role of variables such as \(a^8\) and \(b^7\) is paramount as they must be considered during factorization and simplification.

Applying laws of exponents helps in simplifying these variables under a square root. Break down expressions using rules, for instance, \( \sqrt{a^8} = a^4 \), by recognizing that square roots operate by essentially halving exponents. This understanding is key to resolving radical expressions efficiently.

Radicand Simplification

Simplifying the radicand, which is the expression under the square root sign, is the heart of simplifying radicals.

• The main goal is to express the radicand as simply as possible, ensuring ease of further mathematical operations.
• For complex expressions like \(32a^8b^7\), you start by breaking it down to its simplest form: \( (2^5)(a^8)(b^7) \). Then, you simplify each component individually using previously discussed concepts.

This results in breaking a complex radical into a product of simpler ones, like \(\sqrt{(2^5)(a^8)(b^7)} = 4a^4b^3\sqrt{2b}\), focusing on reducing the powers where possible. Simplifying radicals completely ensures that any operations performed afterward, such as multiplying or adding similar terms, are straightforward and accurate.