Question

Simplify:$\frac{(abc{)}^9\times (a^4{b}^{5}c{)}^{-6}}{(a^{-7}{b}^{-8}{c}^9{)}^4\times (a^{-10}{b}^{11}{c}^{-12}{)}^5\times (a^{13}{b}^{-14}{c}^{15}{)}^2}$

• A. $a^{37}{b}^{16}{c}^3$
• B. $a^{37}{b}^{-16}{c}^{-3}$
• C. $a^{-37}{b}^{-16}{c}^{-3}$
• D. $a^{-37}{b}^{16}{c}^3$

Answer 1

The correct option is B $a^{37}{b}^{-16}{c}^{-3}$

We will be using,
$\to a^m\times a^n=a^{m+n}\text{(Product law)}$
$\to a^m÷a^n=a^{m-n}\text{(Quotient law)}$
$\to (a^m{)}^n=a^{m\times n}\text{(Power to a power law)}$

$\frac{(abc{)}^9\times (a^4{b}^{5}c{)}^{-6}}{(a^{-7}{b}^{-8}{c}^9{)}^4\times (a^{-10}{b}^{11}{c}^{-12}{)}^5\times (a^{13}{b}^{-14}{c}^{15}{)}^2}$

$=\frac{\left(a^9{b}^9{c}^9\right)\times \left(a^{4\times -6}{b}^{5\times -6}{c}^{1\times -6}\right)}{\left(a^{-7\times 4}{b}^{-8\times 4}{c}^{9\times 4}\right)\times \left(a^{-10\times 5}{b}^{11\times 5}{c}^{-12\times 5}\right)\times \left(a^{13\times 2}{b}^{-14\times 2}{c}^{15\times 2}\right)}$

$=\frac{\left(a^9{b}^9{c}^9\right)\times \left(a^{-24}{b}^{-30}{c}^{-6}\right)}{\left(a^{-28}{b}^{-32}{c}^{36}\right)\times \left(a^{-50}{b}^{55}{c}^{-60}\right)\times \left(a^{26}{b}^{-28}{c}^{30}\right)}$

$=\frac{\left(a^9{a}^{-24}\right)\times \left(b^9{b}^{-30}\right)\times \left(c^9{c}^{-6}\right)}{\left(a^{-28}{a}^{-50}{a}^{26}\right)\times \left(b^{-32}{b}^{55}{b}^{-28}\right)\times \left(c^{36}{c}^{-60}{c}^{30}\right)}$

$=\frac{\left(a^{-15}\right)\times \left(b^{-21}\right)\times \left(c^3\right)}{\left(a^{-28-50+26}\right)\times \left(b^{-32+55-28}\right)\times \left(c^{36-60+30}\right)}$

$=\frac{\left(a^{-15}\right)\times \left(b^{-21}\right)\times \left(c^3\right)}{\left(a^{-52}\right)\times \left(b^{-5}\right)\times \left(c^6\right)}$

$=a^{-15-(-52)}{b}^{-21-(-5)}{c}^{3-6}$
$=a^{37}{b}^{-16}{c}^{-3}$