Question

Simplify:

\[\frac{x^2 - 5x - 24}{\left( x + 3 \right)\left( x + 8 \right)} \times \frac{x^2 - 64}{\left( x - 8 \right)^2}\]

Answer 1

It is known that,

a² − b² = (a + b) (a − b)

a³ − b³ = (a − b)(a² + ab + b²)

\[\  \frac{x^2 - 5x - 24}{\left( x + 3 \right)\left( x + 8 \right)} \times \frac{x^2 - 64}{\left( x - 8 \right)^2}\] 

\[ = \frac{x^2 - 8x + 3x - 24}{\left( x + 3 \right)\left( x + 8 \right)} \times \frac{\left( x \right)^2 - \left( 8 \right)^2}{\left( x - 8 \right)^2}\] 

\[ = \frac{x\left( x - 8 \right) + 3\left( x - 8 \right)}{\left( x + 3 \right)\left( x + 8 \right)} \times \frac{\left( x + 8 \right)\left( x - 8 \right)}{\left( x - 8 \right)\left( x - 8 \right)}\] 

\[ = \frac{\left( x + 3 \right)\left( x - 8 \right)}{\left( x + 3 \right)\left( x + 8 \right)} \times \frac{\left( x + 8 \right)\left( x - 8 \right)}{\left( x - 8 \right)\left( x - 8 \right)}\] 

= 1