Problem 36 - Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt[3]{280 a^{5} b^{6}} $$

Step-by-Step Solution

Answer

\(2ab^2\sqrt[3]{35a^2}\) is the simplified form.

Step 1: Identify Perfect Cubes within the Radicand

First, we need to identify any perfect cube factors within the radicand, in this case, 280. The perfect cubes up to 27 are 1, 8, and 27. We can factor 280 as 8 × 35, where 8 is a perfect cube (since 2^3 = 8).

Step 2: Factor Variables into Perfect Cubes

Next, consider the variables in the expression. We have \(a^5\) and \(b^6\). We can express these as \(a^5 = a^3 imes a^2\) and \(b^6 = (b^2)^3\). Both \(a^3\) and \((b^2)^3\) are perfect cubes.

Step 3: Separate Perfect Cubes from Non-Perfect Cubes

Reorganize the expression to separate perfect cubes from non-perfect cubes: \(\sqrt[3]{8 \times 35 \times a^3 \times a^2 \times (b^2)^3}\). We can take the cube roots of \(8\), \(a^3\), and \((b^2)^3\) outside the radical.

Step 4: Take Cube Roots of Perfect Cubes

Calculate the cube roots: \(\sqrt[3]{8} = 2\), \(\sqrt[3]{a^3} = a\), and \(\sqrt[3]{(b^2)^3} = b^2\). Multiply these values outside the radical.

Step 5: Write Simplified Expression

The simplified expression, with perfect cubes factored out, and remaining terms retained inside the radical, becomes: \(2ab^2\sqrt[3]{35a^2}\).

Key Concepts

Perfect Cubes

Perfect cubes are special numbers formed when a number is multiplied by itself two more times. For example, 2 multiplied by itself, then by itself again, gives 8 (i.e., \(2 \times 2 \times 2 = 2^3 = 8\)). This means 8 is a perfect cube. In the exercise, we identified 8 as a perfect cube within the number 280.
Identifying these cubes is crucial because they help simplify radical expressions. By extracting perfect cubes, you can pull them out of a radical sign, making the expression simpler.
To recognize a perfect cube:

• Remember some common cubes: \(1 = 1^3\), \(8 = 2^3\), \(27 = 3^3\), and so on.
• Factor the number to see if any of these cubes are present.
• If you spot a cube, separate it out to simplify the expression.

Simplifying Radicals

Simplifying radicals involves reducing a radical expression to its simplest form. When you encounter a radical, such as a cube root, the goal is to simplify it by removing perfect cubes from under the radical.
Take the given exercise: \(\sqrt[3]{280 a^{5} b^{6}}\). Here’s how to simplify it:

• Identify perfect cube numbers, like 8 in our example from 280.
• Factor variables to seek out cube-like terms, such as \(a^3\) and \((b^2)^3\).
• Separate these cubes from non-perfect cubes: place cube roots outside the radical, and leave leftovers inside.
  For instance: \(\sqrt[3]{8 \times 35 \times a^3 \times a^2 \times (b^2)^3}\) becomes \(2ab^2\sqrt[3]{35a^2}\).

Using these steps makes working with radicals easy and less intimidating.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In the exercise, you needed to simplify a complex algebraic expression involving cube roots and variables \(a\) and \(b\).
Here's how algebraic expressions function:

• They can contain numbers (like 280) and variables with exponents (such as \(a^5\) or \(b^6\)).
• Manipulating these expressions involves using rules of arithmetic and operations like factorization.
• Simplifying them is about making the expression as concise as possible while maintaining its value.

In the context of this exercise, simplifying the algebraic expression means taking cube roots of perfect cubes and simplifying the relationships between the variables and numbers. The result of all this manipulation: \(2ab^2\sqrt[3]{35a^2}\). This shows how being patient with each transformation step results in a simpler form.