Even and Odd Powers: Effects on the Sign
For some values of _x_, the expression $$\frac{x^{a^3}}{x^{ab^2}}$$ is negative. If _a_ and _b_ are integers, which of the following must be true?
To determine how it can be odd, first factor it.
$$\displaystyle a^3-ab^2 = a(a^2 - b^2) = a(a-b)(a+b)$$
This product will be odd only if all three factors, _a_, _a_ – _b_, and _a_ + _b_, are odd. Thus, you know that _a_ must be odd.
If _a_ is odd, then _a_ – _b_ and _a_ + _b_ will only be odd if _b_ is even because an odd plus or minus an even is odd.
Therefore, you should conclude that _a_ must be odd and _b_ must be even.
Incorrect.
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The variable _b_ only appears in the denominator of the expression and there it is raised to the second power, so it doesn't matter what the sign of _b_ is.
To determine how this expression can be odd, first factor it.
$$\displaystyle a^3-ab^2 = a(a^2 - b^2) = a(a-b)(a+b)$$
This product will be odd only if all three factors, $$a$$, $$a-b$$, and $$a+b$$, are odd. Thus, you know that _a_ must be odd.
Incorrect.
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If _a_ is even, then _a_3 and _ab_2 will both be even. Therefore, the exponent $$a^3-ab^2$$ will also be even because an even minus an even is even.
Incorrect.
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Raising a number to a negative exponent flips the numerator and denominator (in other words, takes the reciprocal), so it is not relevant to this problem.
Incorrect.
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If _b_ is odd, then there are two cases:
- If _a_ is even, then _a_3 and _ab_2 are both even, so $$a^3 - ab^2$$ is also even.
- If _a_ is odd, then _a_3 and _ab_2 are both odd, so $$a^3 - ab^2$$ is even again.
In fact, neither case is what you want.
That's correct!
When you divide powers with the same base, subtract the exponent in the denominator from the exponent in the numerator.
$$\displaystyle \frac{x^n}{x^m}=x^{n-m}$$
The given expression is a fraction whose numerator and denominator are both powers whose base is _x_. Thus, you can write it as a single exponent:
$$\displaystyle \frac{x^{a^3}}{x^{ab^2}} = x^{a^3-ab^2}$$.
An even power is always positive or zero, whereas an odd power can be negative, positive, or zero. Thus, in this case, the exponent $$a^3-ab^2$$ must be odd.
That is right!
When you divide powers with the same base, subtract the exponent in the denominator from the exponent in the numerator.
$$\displaystyle \frac{x^n}{x^m}=x^{n-m}$$
The given expression is a fraction whose numerator and denominator are both powers of the same base, _x_. Therefore, you can write it as a single exponent:
$$\displaystyle \frac{x^{a^3}}{x^{ab^2}} = x^{a^3-ab^2}$$
An even power is always positive or zero, whereas an odd power can be anything. So, if the expression is negative for some values of _x_, then $$a^3-ab^2$$ must be odd.
_a_ is positive.
_a_ is even.
_b_ is positive.
_b_ is even.
_b_ is odd.
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