Question

Simplify and express the result in power notation with positive exponent.

1. (−4) 5 ÷ (−4) 8
2. $\left(\frac{1 }{ 2^3}\right)$ 2
3. (−3) 4 × ($\frac{5}{3}$) 4
4. (3 -7 ÷ 3 -10) × 3 -5
5. 2 -3 × (−7) -3

Answer 1

(i) (−4)5 ÷ (−4)8
Since $\frac{a^m}{a^n}$= am − n
(−4)5 ÷ (−4)8 = $\frac{(−4)^5}{(−4)^8}$= (−4)5−8
(−4)−3
$= \frac{1}{(−4)^3}$

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(ii) $\left(\frac{1}{2^3}\right)^2$
Since, (am)n = amn
$⇒ \left(\frac{1}{2^3}\right)^2$
$= \frac{1}{2^6}$

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(iii) (−3)4 $× \left(\frac{5}{3}\right)^4$
We know that am × bm =(ab)m and ($\frac{a}{b}$)m = $\frac{a^m}{b^m}$ where a & b are non-zero integers and m is an integer
$⇒$ (−3)4 $× \left(\frac{5}{3}\right)^4$

$⇒ (−1 × 3)^4 × \frac{5^4}{3^4}$

$⇒$ (−1)4 × 54
= 54 [∵(−1)4 = 1]

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(iv)(3−7 ÷ 3−10) × 3−5
We know. $\frac{a^m}{a^n} =$ am − n and am × an = am + n
(3−7 ÷ 3−10) × 3−5 = (3−7−(−10)) × 3−5
= (3−7 + 10) × 3−5 = 33 × (3−5)
= 33 + (−5)
= 3−2 =$\frac{1}{3^2}$

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(v) 2−3 × (−7)−3
We know that, am × bm = (ab)m
2−3 × (−7)−3
= [2 × (−7)]−3
= (−14)−3
$= \frac{1}{(-14)^3}$ [Since a−m = 1/am]