Answer
1. Evaluate each expression without using a calculator.
a) 81
b)-81
c) $\frac{1}{81}$
d)25
e)$\frac{9}{4}$
f) $\frac{1}{8}$
Work Step by Step
1. Evaluate each expression without using a calculator.
(a) $(-3)^{4}$
Answer: Using: minus one times minus one equals one. ( $-1\times-1 = +1 $) and having $a^{n} = a\times a\times...\times a_{n}$
then:
$ (-3)\times(-3)\times (-3)\times(-3)$ = 81
(b) $-3^{4}$
Answer: $-3^{4} = - (3^{4})$
having $a^{n} = a\times a\times...\times a_{n}$ :
$- ( 3\times3\times3\times3)$ = -81
c) $3^{(-4) }$
Answer: Having: $a^{-1} = \frac{1}{a}$
$\frac{1}{(3\times3\times3\times3)}$= $\frac{1}{81}$
d) $\frac{5^{23}}{5^{21}}$
Answer : Consider: $\frac{a^{b}}{a^{c}} = a^{b-c} $
$5^{2}$= 25
e) $(\frac{2}{3})^{-2}$
Answer: Having: $(\frac{a}{b})^{-1} = \frac{b}{a}$ and considering $(\frac{b}{a})^{n}= \frac{b^{n}}{a^{n}}$
$(\frac{2}{3})^{-2} = \frac{3\times3}{2\times2}$
$\frac{9}{4}$
f)$ 16^{-3/4} $
Answer:
Consider $a^{\frac{b}{c}}= \sqrt[c] (a^{b})$
Having $ \sqrt[4] 16^{- 3} = \sqrt[4] \frac{1}{16\times16\times16} = \sqrt[4] (\frac{1}{16}\times\frac{1}{16}\times\frac{1}{16})$
If we decompose the number 16 we can find the 4th square by finding a number that multipied 4 times is 16. Then:
$16\div2 = 8 $
$8\div2 = 4 $
$4\div2= 2 $
Then we know that $2\times4 = 16 $and that$ \sqrt[4] 16 = 2 $
Considering the expression $\sqrt[a] (\frac{b}{c}) =\frac{\sqrt[a] b}{\sqrt[a] c}$ and having $\sqrt[k] 1 = 1$ (for any k)
$ \sqrt[4] \frac{1}{16\times16\times16} = \sqrt[4] (\frac{1}{16}\times\frac{1}{16}\times\frac{1}{16}) = \frac{1}{2}\times\frac{1}{2}\times\frac{1}{2} = \frac{1}{8}$