Question

Which expression is equivalent to \frac{10q^5w^7}{2w^3}.\ \frac{4\left(q6\right)^2}{w^{-5}} for all values of q and w where the expression is defined?

Answer 1

`(10q^5w^7)/(2w^3).\ (4\left(q^6\right)^2)/(w^(-5)) =20q^(17)w^9`

Explanation:

Given

`(10q^5w^7)/(2w^3).\ (4\left(q^6\right)^2)/(w^(-5))`

Required

Determine the equivalent expression

`(10q^5w^7)/(2w^3).\ (4\left(q^6\right)^2)/(w^(-5))`

Simplify the first fraction

`(5q^5w^7)/(w^3).\ (4\left(q^6\right)^2)/(w^(-5))`

Apply law of indices on the first fraction;

`5q^5w^(7-3).\ (4\left(q^6\right)^2)/(w^(-5))`

`5q^5w^4.\ (4\left(q^6\right)^2)/(w^(-5))`

`\ (5q^5w^4*4\left(q^6\right)^2)/(w^(-5))`

Apply law of indices:

`5q^5w^4*4\left(q^6\right)^2 * w^(5)`

Evaluate the bracket

`5q^5w^4*4 * q^(12) * w^(5)`

Collect Like Terms

`5*4q^5* q^(12)*w^4 * w^(5)`

`20q^5* q^(12)*w^4 * w^(5)`

`20q^(5+12)*w^(4+5)`

`20q^(17)w^9`

Hence:

`(10q^5w^7)/(2w^3).\ (4\left(q^6\right)^2)/(w^(-5)) =20q^(17)w^9`