Question

Which among the following would be positive after simplification?

• A. $(-3{)}^{-7}$
• B. $(-2{)}^{-5}\times (-9{)}^{-7}$
• C. $(-4{)}^4\times (-6{)}^{-3}$
• D. $(-8{)}^{-3}$

Answer 1

The correct option is B $(-2{)}^{-5}\times (-9{)}^{-7}$

$(-2{)}^{-5}\times (-9{)}^{-7}$
$=\left[\right(-1)\times 2{]}^{-5}\times [(-1)\times 9{]}^{-7}$
$=\left[\right(-1{)}^{-5}\times 2^{-5}]\times [(-1{)}^{-7}\times 9^{-7}]$
$(\because (a\times b{)}^n=a^n\times b^n)$

$=(-1{)}^{-5}\times (-1{)}^{-7}\times 2^{-5}\times 9^{-7}$
$=(-1{)}^{-5+(-7)}\times 2^{-5}\times 9^{-7}$
$=(-1{)}^{-12}\times 2^{-5}\times 9^{-7}$
$=\frac{1}{(-1{)}^{12}\times 2^5\times 9^7}$ $(\because a^{-n}=\frac{1}{a^n})$

$=\frac{1}{1\times 2^5\times 9^7}$ $[\because (-1{)}^{12}=1]$

$=\frac{1}{2^5\times 9^7}>0$

Hence, option (b.) is the correct one.