Negative and Zero Exponents: Practice

The product rule adds the exponents, even when one is negative.

$a^6 \cdot a^{-2} = a^{6 + (-2)} = a^{4}$

Flip the fraction, then apply the positive power.

$\left(\frac{1}{3}\right)^{-2} = \left(\frac{3}{1}\right)^2 = 3^2 = 9$

A negative exponent in the denominator becomes a positive exponent in the numerator: $\tfrac{1}{a^{-n}} = a^n$.

$\frac{1}{y^{-4}} = y^4$

A power of a power multiplies the exponents.

$\left(2^{-1}\right)^2 = 2^{(-1)(2)} = 2^{-2} = \frac{1}{4}$

The quotient rule subtracts the exponents, and subtracting a negative adds.

$\frac{5^{2}}{5^{-1}} = 5^{2 - (-1)} = 5^{2+1} = 5^{3}$

Only the factor carrying the negative exponent moves; the coefficient $3$ stays in the numerator.

$3x^{-2} = 3 \cdot \frac{1}{x^2} = \frac{3}{x^2}$

The $3$ does not go to the denominator, since its exponent is $1$, not negative.

Flip the fraction first, then add.

$\left(\frac{1}{5}\right)^{-1} = 5, \qquad 5 + 1 = 6$

Take the reciprocal of $5^2 = 25$, then write the fraction as a decimal.

$5^{-2} = \frac{1}{25} = 0.04$

Subtract the exponents; subtracting a negative adds.

$\frac{x^{-2}}{x^{-5}} = x^{-2 - (-5)} = x^{-2 + 5} = x^{3}$

A negative exponent in the denominator moves to the numerator and turns positive.

$\frac{1}{a^{-6}} = a^6$

A $-1$ exponent inverts the fraction.

$\left(\frac{3}{4}\right)^{-1} = \frac{4}{3}$

Multiply the exponents with the power rule, then rewrite with a positive exponent.

$\left(x^{2}\right)^{-3} = x^{(2)(-3)} = x^{-6} = \frac{1}{x^6}$