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Comparison of Quantities Using Exponents

State whether...

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Question 68

State whether the following statement is true or false:

If x and y are integers such that $x^{2} > y^{2}$ , then $x^{3} > y^{3}$

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Solution

False
Suppose - 1 and - 2 are integers.
Then, $(- 2)^{2} > (- 1)^{2} = 4 > 1$
and $(- 2)^{3} < (- 1)^{3} = - 8 < - 1$

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Question 68

State whether the following statement is true or false:

If x and y are integers such that $x^{2} > y^{2}$ , then $x^{3} > y^{3}$

Question 69

State whether the following statement is true or false:

Let x and y be natural numbers. If x divides y, then $x^{3}$ divides $y^{3}$

Question 69

State whether the following statement is true or false:

Let x and y be natural numbers. If x divides y, then $x^{3}$ divides $y^{3}$

x^{3} + y^{3} + z^{3} - 3 x y z \geq (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x)

x, y \in Z

x

y

x y

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