1. $\left(4^{-1} - 8^{-1}\right) \div \left(\frac{2}{3}\right)^{-1} = \frac{1}{4} - \frac{1}{8} \div \frac{3}{2} = \frac{1}{8} \times \frac{2}{3} = \frac{1}{12}$
2. a) $\left[\left(-\frac{8}{13}\right)^0 + \left(\frac{16}{5}\right)^0\right] \div \left(-\frac{4}{5}\right)^{-1} = (1 + 1) \div \left(-\frac{5}{4}\right) = 2 \times \left(-\frac{4}{5}\right) = -\frac{8}{5}$
b) $\left[\left(-\frac{2}{3}\right)^{-1}\right]^{-1} = \left(-\frac{3}{2}\right)^{-1} = -\frac{2}{3}$
3. Multiplicative inverse of $15^4$ is $15^{-4}$
4. $5^0 \times 3^{-1} = 1 \times \frac{1}{3} = \frac{1}{3}$
5. a) $(-4)^3 \times (-4)^{-8} = (-4)^{-5} = \frac{1}{(-4)^5}$
b) $2^6 \times 2^9 = 2^{15}$
c) $\left(\frac{2}{3}\right)^{-4} \div \left(\frac{2}{3}\right)^5 \times \left(\frac{2}{3}\right)^{13} = \left(\frac{2}{3}\right)^{-4-5+13} = \left(\frac{2}{3}\right)^4$
d) $(-2)^5 \times (-2)^{-7} = (-2)^{-2} = \frac{1}{(-2)^2} = \frac{1}{4}$
6. a) $\frac{4^{-3} \times a^{-5} \times b^{-4}}{4^{-5} \times a^{-4} \times b^3} = 4^{2} \times a^{-1} \times b^{-7} = \frac{16}{a b^7}$
b) $\frac{10^{-4} \times 9^{-4}}{2^{-4} \times 15^{-4}} = \left(\frac{10 \times 9}{2 \times 15}\right)^{-4} = \left(\frac{90}{30}\right)^{-4} = 3^{-4} = \frac{1}{81}$
7. a) $4^{4n-3} = 16^{2n-5} = (4^2)^{2n-5} = 4^{4n-10}$, so $4n - 3 = 4n - 10$ → contradiction; no solution.
b) $2^{2n+3} = 1 = 2^0$, so $2n + 3 = 0$ → $n = -\frac{3}{2}$
8. a) $\left(-\frac{2}{5}\right)^{-2} \div \left(-\frac{2}{5}\right)^{-4} = \left(-\frac{2}{5}\right)^{2} = \frac{4}{25}$, so $\frac{4}{25} = \frac{4}{25}$ → identity, true for all $m$ where defined.
b) $(25)^{-4} = 5^m \Rightarrow (5^2)^{-4} = 5^m \Rightarrow 5^{-8} = 5^m \Rightarrow m = -8$
9. a) $836000000 = 8.36 \times 10^8$
b) $0.000000045 = 4.5 \times 10^{-8}$
c) $0.00000306 = 3.06 \times 10^{-6}$
d) $\frac{6102}{10000} = 0.6102 = 6.102 \times 10^{-1}$
10. a) $6.34 \times 10^{-5} = 0.0000634$
b) $8.9 \times 10^4 = 89000$
c) $6 \times 10^{-4} = 0.0006$
d) $2.3456 \times 10^3 = 2345.6$
Parent Tip: Review the logic above to help your child master the concept of exponents and powers worksheet.
