We're now going to use what we've covered about powers to evaluate some expressions. If you need to, take some time to revise the multiplication law, division law, power of power rule, zero index law, as well as how to deal with negative indeces.
Here are a few things to remember when evaluating powers.
Is $\left(-3\right)^{-3}$(−3)−3 positive, negative or zero?
Because of the negative index law, we know that $\left(-3\right)^{-3}$(−3)−3 will have the same sign as $\left(-3\right)^3$(−3)3, and a negative base raised to an odd power gives a negative result, hence the result will be negative.
We could have figured this out explicitly as follows, but its often much faster to use the previous method of reasoning when answering this type of question.
$\left(-3\right)^{-3}$(−3)−3 | $=$= | $\frac{1}{\left(-3\right)^3}$1(−3)3 |
$=$= | $\frac{1}{\left(-3\right)\times\left(-3\right)\times\left(-3\right)}$1(−3)×(−3)×(−3) | |
$=$= | $\frac{1}{-27}$1−27 | |
$=$= | $-\frac{1}{27}$−127 |
is $-3^{-3}$−3−3 positive, negative or zero?
$-3^{-3}$−3−3 means the negative of whatever $3^{-3}$3−3 is, so we immediately know that its sign will be negative.
We can evaluate $-3^{-3}$−3−3 as follows.
$-3^{-3}$−3−3 | $=$= | $-\frac{1}{3^3}$−133 |
$=$= | $-\frac{1}{3\times3\times3}$−13×3×3 | |
$=$= | $-\frac{1}{27}$−127 |
Evaluate $\left(6\frac{2}{3}\right)^2$(623)2.
To evaluate $\left(6\frac{2}{3}\right)^2$(623)2 we will first need to convert $6\frac{2}{3}$623 to an improper fraction.
$6$6 is equal to $\frac{18}{3}$183 so $6\frac{2}{3}=\frac{18}{3}+\frac{2}{3}$623=183+23 which equals $\frac{20}{3}$203.
Hence, we can evaluate $\left(6\frac{2}{3}\right)^2$(623)2 as follows.
$\left(6\frac{2}{3}\right)^2$(623)2 | $=$= | $\left(\frac{20}{3}\right)^2$(203)2 |
$=$= | $\frac{20^2}{3^2}$20232 | |
$=$= | $\frac{20\times20}{3\times3}$20×203×3 | |
$=$= | $\frac{400}{9}$4009 |
Evaluate $\left(0.012\right)^2$(0.012)2.
$\left(0.012\right)^2=0.012\times0.012$(0.012)2=0.012×0.012 and we can momentarily ignore the decimal point and perform the whole number multiplication $12\times12$12×12 which equals $144$144.
In $0.012$0.012 there is $3$3 decimal places, so adding $3$3 twice tells us that our result should have $6$6 decimal places, so our answer is $0.000144$0.000144.
Is $\left(\frac{1}{4}\right)^{-2}$(14)−2 positive, negative or $0$0?
$0$0
positive
negative
Find the value of $e^{-2.478}$e−2.478 to four decimal places.