Question

Factor the expression 125q^6 - r^6 b^2

Answer 1

`\bf 125q^6-r^6 b^2~~ \begin{cases} 125=&5\cdot 5\\ &5^2\\ q^6=&q^(3\cdot 2)\\ &(q^3)^2\\ r^6=&r^(3\cdot 2)\\ &(r^3)^2 \end{cases}\implies 5^2(q^3)^2-(r^3)^2b^2\implies \stackrel{\textit{difference of squares}}{(5q^3)^2-(r^3b)^2} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill (5q^3-r^3b)(5q^3+r^3 b)~\hfill`

Answer 2

The factorized expression of 125q⁶ - r⁶b² is ((√5)³q³ - r³b)((√5)³q³ + r³b)

How to factorize the expression

From the question, we have the following parameters that can be used in our computation:

125q⁶ - r⁶b²

This can be expressed as

125q⁶ - r⁶b² = 5³q⁶ - r⁶b²

Express 5 as the square of √5

So, we have

125q⁶ - r⁶b² = [(√5)²]³q⁶ - r⁶b²

Express as squares, we have

125q⁶ - r⁶b² = ((√5)³q³)² - (r³b)²

Applying the difference of two squares, we have

125q⁶ - r⁶b² = ((√5)³q³ - r³b)((√5)³q³ + r³b)

Hence, the factorized expression is ((√5)³q³ - r³b)((√5)³q³ + r³b)