Integer Exponents - Pre Algebra

1. What does a negative exponent mean, as in $a^{-n}$?

   $a^{-n} = \frac{1}{a^n}$ for $a \ne 0$. It means the reciprocal of the positive power.

2. What is the value of $2^{-3}$?

   $\frac{1}{8}$. Compute $\frac{1}{2^3}$.

3. What is the value of $5^{-2}$?

   $\frac{1}{25}$. Compute $\frac{1}{5^2}$.

4. What is the value of $\frac{1}{4^{-2}}$?

   $16$. Since $4^{-2} = \frac{1}{16}$, its reciprocal is $16$.

5. What is the value of $(-2)^3$?

   $-8$. Compute $(-2)(-2)(-2)$; an odd number of negative factors gives a negative result.

6. What is the value of $10^{-3}$ as a decimal?

   $0.001$. It equals $\frac{1}{1000}$.

7. What is the value of $3^{-1} \cdot 3^3$?

   $9$. Add the exponents: $3^{-1+3} = 3^2$.

8. What is the value of $\left(\frac{1}{2}\right)^{-2}$?

   $4$. The reciprocal of $\frac{1}{2}$ is $2$, and $2^2 = 4$.

9. What is the value of $(-3)^2$?

   $9$. The base is $-3$, and $(-3)(-3) = 9$.

10. What is the value of $a^0$ for any nonzero number $a$?

    $1$. For example, $9^0 = 1$.

11. When a negative base is raised to an even whole-number exponent, what is the sign of the result?

    Positive. The negative factors pair up, and each pair multiplies to a positive number.

12. What is the value of $-3^2$?

    $-9$. The exponent applies only to $3$, so this is $-(3 \cdot 3)$.

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