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