Exponents with Integer and Rational Bases

When working with exponents, the base is the number being multiplied, and the exponent tells you how many times to multiply it by itself. Things can get a little tricky when the base is a negative integer or a fraction (rational number).

The Crucial Difference: $(-a)^n$ vs $-a^n$

The most common mistake with exponents involves negative signs. Parentheses dictate exactly what is being raised to the power.

1. Negative Inside Parentheses: $(-a)^n$ When the negative sign is inside the parentheses, it is part of the base. You multiply the entire negative number by itself.

• Even powers result in a positive number because pairs of negatives cancel out. $(-3)^2 = (-3) \times (-3) = 9$
• Odd powers result in a negative number because one negative sign will always be left over. $(-2)^3 = (-2) \times (-2) \times (-2) = -8$

2. Negative Outside (No Parentheses): $-a^n$ When there are no parentheses, the exponent applies only to the number immediately before it, not the negative sign. You calculate the power first, then make the final result negative.

• Example: $-5^2 = -(5 \times 5) = -25$

(Notice how $-5^2$ equals $-25$, but if we had written $(-5)^2$, the answer would be $25$.)

Rational Bases (Fractions)

When raising a fraction to a power, you apply the exponent to both the numerator (the top number) and the denominator (the bottom number).

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Example: $\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}$

If the fraction is negative, the exact same even/odd power rules apply as they do with integers. Since the negative is inside the parentheses and the power is odd, the result stays negative: $\left(-\frac{1}{2}\right)^3 = \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) = -\frac{1}{8}$