Question

${\left(\frac{5^a}{5^b}\right)}^{a+b}.{\left(\frac{5^b}{5^c}\right)}^{b+c}.{\left(\frac{5^c}{5^a}\right)}^{c+a}=$

• A. $1$
• B. $4$
• C. $5$
• D. $0$

Answer 1

The correct option is A $1$

We have, $(\frac{5^a}{5^b}{)}^{a+b}\cdot (\frac{5^b}{5^c}{)}^{b+c}\cdot (\frac{5^c}{5^a}{)}^{c+a}$

$=(5^{a-b}{)}^{a+b}\cdot (5^{b-c}{)}^{b+c}\cdot (5^{c-a}{)}^{c+a}$

$=5^{a^2-b^2}\cdot 5^{b^2-c^2}\cdot 5^{c^2-a^2}$

$=\frac{5^{a^2}}{5^{b^2}}\cdot \frac{5^{b^2}}{5^{c^2}}\cdot \frac{5^{c^2}}{5^{a^2}}=1$