# How do you multiply (25a^2b)^3(1/5abc)^2?

Apr 22, 2015

Start with the exponent rule ${\left({x}^{m}\right)}^{n} = {x}^{m \cdot n}$ .

${\left(25 {a}^{2} b\right)}^{3} {\left(\frac{1}{5} a b c\right)}^{2}$=

$\left({25}^{3} {a}^{6} {b}^{3}\right) \left({\left(\frac{1}{5}\right)}^{2} {a}^{2} {b}^{2} {c}^{2}\right)$=

$\left(15625 {a}^{6} {b}^{3}\right) \left(\left(\frac{1}{25}\right) {a}^{2} {b}^{2} {c}^{2}\right)$=

Divide 15625 by 25 and use the exponent rule $\left({x}^{m}\right) \left({x}^{n}\right) = {x}^{m + n}$.

$\left(\frac{15625}{25}\right) \left({a}^{6 + 2}\right) \left({b}^{3 + 2}\right) \left({c}^{2}\right) = 625 {a}^{8} {b}^{5} {c}^{2}$

Apr 22, 2015

We know that color(blue)((ab)^2 = a^2 * b ^2

Hence
${\left(25 {a}^{2} b\right)}^{3} {\left(\frac{1}{5} a b c\right)}^{2}$

$= \left\{{25}^{3} \cdot {\left({a}^{2}\right)}^{3} \cdot {b}^{3}\right\} \cdot \left\{{\left(\frac{1}{5}\right)}^{2} \cdot {a}^{2} \cdot {b}^{2} \cdot {c}^{2}\right\}$

$= \left\{{\left({5}^{2}\right)}^{3} \cdot {a}^{6} \cdot {b}^{3}\right\} \cdot \left\{{\left(\frac{1}{5}\right)}^{2} \cdot {a}^{2} \cdot {b}^{2} \cdot {c}^{2}\right\}$

$= \left\{{5}^{6} \cdot {a}^{6} \cdot {b}^{3}\right\} \cdot \left\{{\left(\frac{1}{5}\right)}^{2} \cdot {a}^{2} \cdot {b}^{2} \cdot {c}^{2}\right\}$

Next, we group the Constants and the Same Variables together

$= \left({5}^{6} \cdot {\left(\frac{1}{5}\right)}^{2}\right) \cdot \left({a}^{6} \cdot {a}^{2}\right) \cdot \left({b}^{3} \cdot {b}^{2}\right) \cdot {c}^{2}$

$= \left({5}^{6} / {5}^{2}\right) \cdot \left({a}^{6} \cdot {a}^{2}\right) \cdot \left({b}^{3} \cdot {b}^{2}\right) \cdot {c}^{2}$

Two important laws of Exponents:

color(green)(a^m*a^n = a^(m+n) if a!=0
color(green)(a^m/a^n = a^(m -n) if a!=0

Applying these, we get

$= \left({5}^{6 - 2}\right) \left({a}^{6 + 2}\right) \cdot \left({b}^{3 + 2}\right) \cdot {c}^{2}$

$= {5}^{4} \cdot {a}^{8} \cdot {b}^{5} \cdot {c}^{2}$

$= 625 {a}^{8} {b}^{5} {c}^{2}$