Question

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

Answer 1

Start with the exponent rule `(x^m)^n=x^(m*n)` .

`(25a^2b)^3(1/5abc)^2`=

`(25^3a^6b^3)((1/5)^2a^2b^2c^2)`=

`(15625a^6b^3)((1/25)a^2b^2c^2)`=

Divide 15625 by 25 and use the exponent rule `(x^m)(x^n)=x^(m+n)`.

`(15625/25)(a^(6+2))(b^(3+2))(c^2)=625a^8b^5c^2`

Answer 2

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

Hence
`(25a^2b)^3(1/5abc)^2`

`= {25^3 * (a^2)^3 * b^3} * {(1/5)^2 * a^2 * b^2 * c^2}`

`= {(5^2)^3 * a^6 * b^3} * {(1/5)^2 * a^2 * b^2 * c^2}`

`= {5^6 * a^6 * b^3} * {(1/5)^2 * a^2 * b^2 * c^2}`

Next, we group the Constants and the Same Variables together

`= (5^6*(1/5)^2)*(a^6*a^2)*(b^3*b^2)*c^2`

`= (5^6/5^2)*(a^6*a^2)*(b^3*b^2)*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

`= (5^(6-2))(a^(6+2))*(b^(3+2))*c^2`

`= 5^4 * a^8 * b^5 *c^2`

`= 625a^8b^5c^2`