Questions: 27u^3 - 125

Transcript text: \[ 27 u^{3}-125 \]

Solution

Solution Steps

Step 1: Identify the Expression as a Difference of Cubes

The given expression is: \[ 27u^3 - 125 \] We recognize this as a difference of cubes, which can be factored using the formula: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

Step 2: Determine \(a\) and \(b\)

To apply the formula, we need to find \(a\) and \(b\) such that: \[ a^3 = 27u^3 \quad \text{and} \quad b^3 = 125 \] Thus, we have: \[ a = \sqrt[3]{27u^3} = 3u \quad \text{and} \quad b = \sqrt[3]{125} = 5 \]

Step 3: Apply the Difference of Cubes Formula

Substitute \(a = 3u\) and \(b = 5\) into the difference of cubes formula: \[ 27u^3 - 125 = (3u - 5)((3u)^2 + (3u)(5) + 5^2) \]

Step 4: Simplify the Expression

Simplify the terms inside the parentheses: \[ (3u)^2 = 9u^2, \quad (3u)(5) = 15u, \quad 5^2 = 25 \] Thus, we have: \[ 27u^3 - 125 = (3u - 5)(9u^2 + 15u + 25) \]

Final Answer

\[ \boxed{(3u - 5)(9u^2 + 15u + 25)} \]

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