# Question #9e52a

Sep 30, 2016

The product rule and the power rules in question are

• ${a}^{m} \times {a}^{n} = {a}^{m + n}$

• ${a}^{m} \div {a}^{n} = {a}^{m - n}$

• ${\left({a}^{m}\right)}^{n} = {a}^{m \times n}$

(Note that the second rule comes directly from the first, because $\frac{1}{a} ^ x = {a}^{- x}$, so ${a}^{m} / {a}^{n} = {a}^{m} \times {a}^{- n} = {a}^{m + \left(- n\right)} = {a}^{m - n}$)

Rather than simply post answers to all of the questions on the worksheet, here are a couple of examples of each. These should show how to do the rest, as the problems are all effectively the same, save for the changed values.

First set:

1) ${5}^{- 8} \times {5}^{- 5} = {5}^{- 8 + \left(- 5\right)} = {5}^{- 13}$

2) ${18}^{- 4} \times {18}^{3} = {18}^{- 4 + 3} = {18}^{- 1}$

Second set:

1) ${4}^{2} \div {4}^{10} = {4}^{2 - 10} = {4}^{- 8}$

2) ${19}^{7} \div {19}^{- 8} = {19}^{7 - \left(- 8\right)} = {19}^{15}$

Third set:

1)${\left({15}^{9}\right)}^{- 7} = {15}^{9 \times - 7} = {15}^{- 63}$

2)${\left({7}^{3}\right)}^{6} = {7}^{3 \times 6} = {7}^{18}$