Question

Find the value of m in each equation below. Justify your answer.
(x^m × x^2)^3 (k^3)^5 = x^21k^15

Answer 1

The main properties we use in this problem are:

i) $x^a \cdot x^b=x^{a+b}$ (so, when we multiply 2 exponents of the same base, we add the exponents)

ii) $({x^a})^b=x^{a \cdot b}=x^{ab}$


thus, 

$(x^m \cdot x^2)^3=({x^{2m}})^3=x^{2m \cdot3}=x^{6m}$

by first applying property i) then property ii)

also, $({k}^3)^5=k^{15}$, by property ii)



So we have:

$({x^m \cdot x^2})^3 \cdot({k}^3)^5=x^{21} \cdot k^{15}\\\\x^{6m}\cdot k^{15}=x^{21} \cdot k^{15}$, 


now we only have to compare the exponents. 

6m must be equal to 21, 

thus m=21/6=7/2


Answer: m=7/2