Question

Solve the given algebraic expression: \[\dfrac{{{x}^{2}}-5x-24}{\left( x+3 \right)\left( x+8 \right)}\times \dfrac{{{x}^{2}}-64}{{{\left( x-8 \right)}^{2}}}\]

Answer 1

Hint: To solve this type of expression\[\dfrac{{{x}^{2}}-5x-24}{\left( x+3 \right)\left( x+8 \right)}\times \dfrac{{{x}^{2}}-64}{{{\left( x-8 \right)}^{2}}}\] we have to use the factorization method to obtain the factors and cancelling the terms which are in common gives the final solution of the given expression. All the common factors which are in numerator and denominator are cancelled to obtain the solution.

Complete step-by-step solution -
The given expression is\[\dfrac{{{x}^{2}}-5x-24}{\left( x+3 \right)\left( x+8 \right)}\times \dfrac{{{x}^{2}}-64}{{{\left( x-8 \right)}^{2}}}\]
 In the numerator there is term\[{{x}^{2}}-5x-24\]. . . . . . . . . . . . . . . . . . . . (1)
We have to factorize the given term.
The factorization is shown as below
\[{{x}^{2}}-8x+3x-24\]
\[x\left( x-8 \right)+3\left( x-8 \right)\]
\[\left( x-8 \right)\left( x+3 \right)\]. . . . . . . . . . . . . . . . . . . . . . . . . . .(2)
For expression (1) the factors are shown in (2)
Now writing the complete expression as\[\dfrac{\left( x+3 \right)\left( x-8 \right)}{\left( x+3 \right)\left( x+8 \right)}\times \dfrac{{{x}^{2}}-64}{{{\left( x-8 \right)}^{2}}}\]
Now taking the other term in the expression\[{{x}^{2}}-64\] and factorizing it, the solution obtained is \[\left( x+8 \right)\left( x-8 \right)\]
Now writing the complete expression as\[\dfrac{\left( x+3 \right)\left( x-8 \right)}{\left( x+3 \right)\left( x+8 \right)}\times \dfrac{\left( x+8 \right)\left( x-8 \right)}{{{\left( x-8 \right)}^{2}}}\]
We can see the common terms in the given expression \[\dfrac{\left( x+3 \right)\left( x-8 \right)}{\left( x+3 \right)\left( x+8 \right)}\times \dfrac{\left( x+8 \right)\left( x-8 \right)}{{{\left( x-8 \right)}^{2}}}\]
We have to cancel all the common terms in expression therefore the solution obtained is as follows, \[1\]
  Therefore we can conclude that for the given expression
\[\dfrac{{{x}^{2}}-5x-24}{\left( x+3 \right)\left( x+8 \right)}\times \dfrac{{{x}^{2}}-64}{{{\left( x-8 \right)}^{2}}}\]= \[\dfrac{\left( x+3 \right)\left( x-8 \right)}{\left( x+3 \right)\left( x+8 \right)}\times \dfrac{\left( x+8 \right)\left( x-8 \right)}{{{\left( x-8 \right)}^{2}}}\]
The solution is 1.

Note: This is a direct problem with the factors in both numerator and denominator, by cancelling all the terms we get the solution. we should take care while factorizing the expression as that is the key step in the solution. The factors can be obtained by using a formula or by using trial and error methods.