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Long Division

Write true T ...

Question

Write true $(T)$ or false $(F)$ for the following statements:
If $a$ and $b$ are integers such that $a^{2} > b^{2},$ then $a^{3} > b^{3}$

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Solution

Step: State true or false

Given statement: If $a$ and $b$ are integers such that $a^{2} > b^{2},$ then $a^{3} > b^{3}$

Consider the values: $a = - 5$ and $b = - 4$
$\Rightarrow a^{2} = (- 5)^{2}$
$= 25$

$\Rightarrow b^{2} = (- 4)^{2}$
$= 16$

$\Rightarrow a^{3} = (- 5)^{3}$
$= - 125$

$\Rightarrow b^{3} = (- 4)^{3}$
$= - 64$

Clearly, $a^{2} > b^{2}$ but $a^{3} ≯ b^{3}$

Therefore, the given statement is not true for all values of $a$ and $b .$

Hence, the given statement is false.

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