Question

How do you simplify `x^2*x^sqrt3`?

Answer 1

`x^2*x^sqrt3=x^(2+sqrt3)`

Explanation:

Imagine, first, we wanted to simplify the expression `x^2*x^3`. Recall that a power of `2` simply means "multiplied by itself twice."

Then,

`x^2*x^3=overbrace(x*x)^(x^2)*overbrace(x*x*x)^(x^3)=overbrace(x*x*x*x*x)^(x^5)=x^5`

In general, we can write that:

`x^a*x^b=overbrace(x*x*...*x)^("x multiplied a times")*overbrace(x*x*...*x)^("x multiplied b times")=overbrace(x*x*...*x)^("x multiplied a+b times")=x^(a+b)`

In the given problem, one of the powers is `sqrt3`, an irrational number. However, this has no bearing on the above rule `x^a*x^b=x^(a+b)`.

`x^2*x^sqrt3=x^(2+sqrt3)`